Alphabet & Epstein Numbers

The third phase of Counting really,Really,REALLY High

A power-tower of a Zurvanic Absolute Number POPBLED the Zurvanic Absolute Number popbled the Zurvanic Absolute Number times, times
Zurvanic Absolute Numbers,
each surrounded by Moser polygons of a number equal to
the Zurvanic Absolute Number times the Zurvanic Absolute Number power of its upward-counting ordinal as every term in a Zurvanic-Absolute-Number-per-dimension, Zurvanic Absolute Number dimensional Bowers array,
each polygon having a number of sides equal to the number of Zurvanic Absolute Numbers,raised to itself the-Zurvanic-Absolute-Number-popbled Zurvanic Absolute Numbers of times, popbled a number of times equal to the Zurvanic Absolute Number in a Zurvanic-Absolute-Number-in-a-Zurvanic-Absolute-Number-gon-gon times a power tower of the number of polygons surrounding that particular Zurvanic Absolute Number raised to the Zurvanic Absolute Number times its Zurvanic Absolute Number power in a number of layers equal to the Zurvanic Absolute Number times the Zurvanic Absolute Number power of the Zurvanic Absolute Number times itself-raised-to-the-Zurvanic-Absolute-Number-times-its-Zurvanic-Absolute-Number-in-a-Zurvanic-Absolute-Number-gon-power,
the whole popbled the Zurvanic Absolute Number times the number of times expressed by a chain of Conway arrows between every number rising from twenty-nine to the Zurvanic Absolute Number power of the Zurvanic Absolute Number power of the Zurvanic Absolute Number, plus the Zurvanic Absolute Number ultrexed the Zurvanic Absolute Number of times to the Zurvanic Absolute Number, each number repeated the Zurvanic Absolute Number of times,
is defined as "the Number a", first of The Alphabet Numbers.

The convoluted collection of operations described above is the Alphabet Number function.
Perform the Alphabet Number Function on The Number a (substituted in each instance for the Zurvanic Absolute Number) and repeat a times,and you reach the Number A,perform it on A repeating A times and you reach The Number b, and so on through the alphabet.
Performing the function on the Number Z that many times yields the Number aa,followed in sequence by aA,ab,and so forth...after aZ comes Aa,after ZZ comes aaa,and so on.

In this terminology,we quickly realize that TWO plus TWO is a lot less than FOUR, which is more than SIX but much less than ZERO,while every number discussed so far is a great deal less than NOTHING.
We also discover that SOMETHING is much greater than INFINITY even though it doesn't come close to CHUMPCHANGE.What's more,BIGGERTHANINFINITY is way less than MUCHTOOSMALLTOMENTION.
And we realize that we need even more methods of conveying numbers much larger.

Bracket colon notation

The Number expressed by the Number Z Z's is expressed in shorthand as
{Z:Z}
and that expressed by THAT many Zs is
{Z:Z:Z}
while the number expressed by repeating the single-colon-separated Zs Z times is
{Z::Z}
.

The number expressed by THAT many Zs is
{Z:Z::Z}
while the number expressed by using that many colons to separate Zs is
{Z[Z::Z]Z}
Alphabet Numbers separated by N colons convert to
{(first alphabet number)((N-1 colons,second alphabet number -1) repeated second alphabet number -1 times)}

To convey using that many layers of square brackets use
{Z{[Z::Z]Z}}.
Brackets resolve from inside to out and right to left.
{ONE:TWO:THREE:FOUR}
means FOUR repetitions of THREE is the number of repetitions of TWO needed to express the number of repetitions of ONE to express the intended number,while
{ONE[TWO:THREE]FOUR}
means THREE repetitions of TWO is the number of colons to calculate with in repeating FOUR repetitions of FOUR (working leftward and repeating FOUR each successively calculated number of times) to get the number of repetitions of ONE.
The rightmost bracket closes all unclosed pairs,if any are outstanding.

Epstein's First Number

With this groundwork laid,it is finally possible to express numbers big enough to put my name on.

The Number

{GREATERTHANYOUCOULDPOSSIBLYDESCRIBEORIMAGINE
(GREATERTHANYOUCOULDPOSSIBLYDESCRIBEORIMAGINE colons)
{[GREATERTHANYOUCOULDPOSSIBLYDESCRIBEORIMAGINE
(GREATERTHANYOUCOULDPOSSIBLYDESCRIBEORIMAGINE colons)
{[GREATERTHANYOUCOULDPOSSIBLYDESCRIBEORIMAGINE
(GREATERTHANYOUCOULDPOSSIBLYDESCRIBEORIMAGINE colons)
.
.
.
(repeated GREATERTHANYOUCOULDPOSSIBLYDESCRIBEORIMAGINE! layers)
.
.
.
ISAIDGREATERTHANYOUCOULDPOSSIBLYDESCRIBEORIMAGINE!}
Zs express Epstein's First Number,or E₍₁₎... in full "E_{(1,0,0,0,0...)}" as explained hereafter.

The Epstein Number Function

E_(n) is graduated to E_(n+1) as follows.

1.Popble E_(n) E_(n) times.
2.Perform the Alphabet Number Function on the number resulting from Step 1,that many times.
3.The Number expressed by the number resulting from Step 2 Zs is then substituted for GREATERTHANYOUCOULDPOSSIBLYDESCRIBEORIMAGINE in every appearance in the expression defining E₍₁₎ (retaining the two factorials),and this is repeated that many times.
4.Steps 1-3 are repeated E_(n) times.

E(E(E(E(E(...E(n) layers...(n))))))
is also E_(n,1).
E(E(E(E(E(...E(n,1) layers...(n,1))))))
is also E_(n,2).
E(E(E(E(E(...E(E(n),E(n)layers) layers...(E(n),E(n)))))))
is also E_(n,1,1).
E(E(E(E(E(...E(E(E(n),E(n),E(n)),(E(n),E(n),E(n) layers)layers) layers...(E(n),E(n),E(n))))))))
is also E_(n,1,1,1).
etc.

The Nominated Epstein Numbers

Epstein's First (Unofficially) NOMINATED Number

And so,the first number I will use in offering a nomination for biggest named number is
E_{(I,SAID,GREATER,THAN,ANY,OF,YOU,COULD,POSSIBLY,DESCRIBE,OR,IMAGINE!,NOT,IN,SEVENTY,{OCTO:YOTTILLIARD},TIMES,AS,MANY,FORTNIGHTS,AS,THERE,HAVE,BEEN,PLANCK,TIMES,SINCE,THE,BIG,BANG!)}

Of course that's so easily beatable...

Epstein's SECOND (Unofficially) Nominated Number

E_{(THIS,NUMBER,IS,SO,MUCH,BIGGER,THAN,MY,FIRST,NOMINATED,NUMBER,IT,WILL,BOGGLE,ALL,OF,YOUR,MINDS!,SEVENTEEN,{OCTO:YOTTILLIO:ILLIARDS},OF,BUSY,BEAVERS,COULD,NEVER,EVER,COME,ANYWHERE,NEAR,TO,THIS,HUUGE!...repeated
Epstein's First Nominated
Number,popbled,times...I,MEAN,THIS,NUMBER,IS,TRULY,GARGANTUANLY,TREMENDOUS!)}

Epstein's THIRD (Unofficially) Nominated Number

My third nominated number is

the Second Nominated Number with
the Number expressed by
the Second Nominated Number,with the Alphabet Number function performed upon it
the number of times expressed by a chain of Conway arrows between ten repetitions of every number from five to itself ultrexed twice to its factorial,
Zs repetitions
of this number of repetitions
interpolated after the first "THIS",and E_{(that number of repetitions)} repetitions of that number interpolated after the first "NUMBER",and the number of layers of E-subscripting before the immediately preceding number is used to to get the number and its number of repetitions raised to itself in a power-tower of the immediately preceding number of times after every succeeding array generation(Alphabet Number in the specification).

This is the embryo of the Epstein Number Nominating Function.
From this point,to get from Unofficially Nominated Number n to Unofficially Nominated Number n+1,a similar interpolation of additional generations is performed as follows.

1.After the first specified generation in EUNN n,
add the Alphabet Number described by
EUNN n with the Alphabet Number Function performed upon it
E_{(1,2,3,...E_{(EUNN n)})} times,
Z's,
repetitions of that Alphabet Number.
2.After each succeeding generation,interpolate
E_{E(...number of layers of subscripting equal to number interpolated
after preceding generation raised to itself in a power tower of a number
of layers equal to itself)_{(number interpolated in previous
generation)})}) repetitions of this new number of repetitions.

3.Repeat steps 1 and 2,substituting the entire number achieved after step 2 for EUNN n in step 1,
E_{(1,2,3,4,...E_{(E_{(EUNN n layers of subscripting)(EUNN
n)})})} times.

When EUNN n reaches E_{(1,2,3,...E_{E_{(Epstein's Third
Unofficially Nominated number of layers of subscripts)(Epstein's Third
Unofficially Nominated Number)})})} we have reached Epstein's One Step Toward A Little Bit Seriously Nominated Number.

The increment in EUNN n to take ESTALBSNN n to n + 1 is the
E_{(1,1,2,2,3,3...E_{E_{(ESTALBSNN n layers of
subscripts)(ESTALBSNN
n,repeated ESTALBSNN n times)})})} power of the previous number.

When ESTALBSNN n reaches E_{((every integer
from 1 repeated twice its square times)...E_{(E_{(E_{Epstein's
Fourth Step Toward A Little Bit Seriously Nominated Number)}
layers of subscripts)(Epstein's Fourth Step Toward a Little Bit Seriously
Nominated Number,repeated E_{(Epstein's Fourth Step Toward a Little Bit
Seriously Nominated Number)} times)})})} we have reached Epstein's First A Little Bit Seriously Nominated Number.

The increment in ESTALBSNN n to take EALBSNN n to n+1 is a power tower of the previous increment raised to E_{((every integer from 1 repeated thrice its cube times)...E(E(EALBSNN
n,repeated E(EALBSNN n layers of subscripts) layers of
subscripts)(EALBSNN n),
repeated that many times)))}) that many times.

When EALBSNN n reaches E_{((every integer
from 1 repeated four times its fourth power
times)...E_{(E_{(E_{Epstein's
Fifth A Little Bit Seriously Nominated Number,repeated
Epstein's Fifth A Little Bit Seriously Nominated Number times
)} layers of subscripts)(E(Epstein's Fifth A Little Bit Seriously
Nominated Number,repeated Epstein's Fifth A Little Bit Seriously Nominated
Number times),repeated E_{(Epstein's Fifth A Little Bit
Seriously Nominated Number)} times)})})} we have reached Epstein's First Step Toward Semi-Seriously Nominated Number.

The increment in EALBSNN n to take ESTSSNN n to n + 1 is a power tower of E_{(the previous increment raised to
E_{((every integer from 1 repeated four times its fifth power
times)...E(E(ESTSSNN
n,repeated E(E(ESTSSNN n,repeated ESTSSNN n times) layers of subscripts)
layers of subscripts)(ESTSSNN n),
repeated that many times)))}) that many times)} raised to itself that many times.

When ESTSSNN n reaches E_{E_{((every integer
from 1 repeated four times its sixth power
times)...E_{(E_{(E_{(Epstein's
Sixth Step Toward Semi-Seriously Nominated Number,repeated
E(Epstein's Sixth Step Toward Semi-Seriously Nominated Number) times
)} layers of subscripts)(E(Epstein's Sixth Step Toward Semi-Seriously
Nominated Number,repeated E(Epstein's Sixth Step Toward Semi-Seriously
Nominated Number) times),repeated E_{(Epstein's Sixth Step Toward
Semi-Seriously Nominated Number)} times)})})})} we have reached Epstein's First Semi-Seriously Nominated Number.

The increment in ESTSSNN n to take ESSNN n to n + 1 is a power tower of E_{(the previous increment raised to
E_{((every integer from 1 repeated four times its seventh power
times)...E(E(ESSNN n,repeated E(E(ESSNN n,repeated E(E(ESSNN
n,repeated ESSNN n times) times) layers of subscripts)
layers of subscripts)(E(ESSNN n,repeated ESSNN n times)),
repeated that many times)))}) that many times)} raised to itself E(that many times) times.

When ESSNN n reaches E_{E_{((every integer
from 1 repeated four times its eighth power
times)...E_{(E_{(E_{(Epstein's
Seventh Semi-Seriously Nominated Number,repeated
E(Epstein's Seventh Semi-Seriously Nominated Number) times
)} layers of subscripts)(E(Epstein's Seventh Semi-Seriously
Nominated Number,repeated E(Epstein's Seventh Semi-Seriously
Nominated Number) times),repeated E_{(Epstein's Seventh
Semi-Seriously Nominated Number)}
times)})}),repeated E(that many times) times})} we have reached Epstein's First Step Toward Almost Seriously Nominated Number.

The increment in ESSNN n to take ESTASNN n to n + 1 is a power tower of E_{(the previous increment raised to
E_{((every integer from 1 repeated four times its ninth power
times)...E(E(ESTASNN n,repeated E(E(ESTASNN n,repeated E(E(ESTASNN
n,repeated E(ESTASNN n,repeated ESTASNN n times) times) times) layers of
subscripts)
layers of subscripts)(E(ESTASNN n,repeated E(ESTASNN n) times)),
repeated that many times)))}) that many times)} raised to itself E(that many times,repeated E_{(that many times)} times) times.

When ESTASNN n reaches E_{E_{((every integer
from 1 repeated five times its tenth power
times)...E_{(E_{(E_{(Epstein's
Eighth Step Toward Almost Seriously Nominated Number,repeated
E(Epstein's Eighth Step Toward AlmostSeriously Nominated Number) times
)} layers of subscripts)(E(Epstein's Eighth Step Toward
Almost Seriously Nominated Number,repeated E(Epstein's Eighth
Step Toward Almost Seriously Nominated Number) times),repeated E_{(Epstein's Eighth Step Toward Almost Seriously Nominated Number,
repeated E(Epstein's Eighth Step Toward Almost Seriously Nominated
Number) times)}
times)})}),repeated E(that many times,repeated E(that many
times)times ) times})} we have reached Epstein's First Almost Seriously Nominated Number.

The increment in ESTASNN n to take EASNN n to n + 1 is a power tower of E_{(the previous increment raised to
E_{((every integer from 1 repeated six times its twelfth power
times)...E(E(EASNN n,repeated E(E(EASNN n,repeated E(E(EASNN
n,repeated E(EASNN n,repeated E(EASNNN n,repeated EASNN n times) times)
times) times) layers of
subscripts)
layers of subscripts)(E(EASNN n,repeated E(EASNN n,repeated EASNN n times)
times)),
repeated that many times)))}) that many times)} raised to itself E(that many times,repeated E(E_{(that many times)}) times) times.

When EASNN n reaches E_{E_{((every integer
from 1 repeated ten times its twentieth power
times)...E_{(E_{(E_{(Epstein's
Ninth Almost Seriously Nominated Number,repeated
E(Epstein's Ninth Almost Seriously Nominated Number,repeated
E(Epstein's Ninth Almost Seriously Nominated Number) times) times
)} layers of subscripts)(E(Epstein's Ninth
Almost Seriously Nominated Number,repeated E(Epstein's Ninth
Almost Seriously Nominated Number) times),repeated E_{(Epstein's Ninth Almost Seriously Nominated Number,
repeated E(Epstein's Ninth Almost Seriously Nominated
Number) times)}
times)})}),repeated E(that many times,repeated E(that many
times)times ) times})} we have reached Epstein's First Step Toward Seriously Nominated Number.

The increment in EASNN n to take ESTSNN n to n + 1 is a power tower of E_{(the previous increment raised to
E_{((every integer from 1 repeated ten times its twenty-fifth power
popbled times)...E(E(ESTSNN n,repeated E(E(ESTSNN n,repeated E(E(ESTSNN
n,repeated E(ESTSNN n,repeated E(ESTSNN n,repeated E(ESTSNN n) times)
times)
times) times) layers of subscripts)
layers of subscripts)(E(ESTSNN n,repeated E(ESTSNN n,repeated E(ESTSNN n)
times) times)),
repeated that many times)))}) that many times)} raised to itself E(E_{(that many times)},repeated E(E_{(E(that many
times))}) times) times.

When ESTSNN n reaches E_{E_{((every integer
from 1 repeated twenty times its thirtieth power popbled
times)...E_{(E_{(E_{(Epstein's
Tenth Step Toward Seriously Nominated Number,repeated
E(Epstein's Tenth Step Toward Seriously Nominated Number,repeated
E(Epstein's Tenth Step Toward Seriously Nominated Number) times) times
)} layers of subscripts)(E(Epstein's Tenth Step
Toward Seriously Nominated Number,repeated E(Epstein's Tenth Step
Toward Seriously Nominated Number) times),repeated E_{(Epstein's Tenth Step Toward Seriously Nominated Number,
repeated E(Epstein's Tenth Step Toward Seriously Nominated
Number,repeated E(Epstein's Tenth Step Toward Seriously Nominated
Number) times) times)}
times)})}),repeated E(that many times,repeated E(that many
times)times ) times})} we have at last reached Epstein's First Seriously Nominated Number.

The increment in ESTSNN n to take ESNN n to n + 1 is a power tower of E_{(the previous increment raised to
E_{((every integer from 1 repeated twenty-two times its fortieth
power,popbled twice,times)...E(E(ESNN n,repeated E(E(ESNN n,repeated
E(E(ESNN
n,repeated E(ESNN n,repeated E(E(ESNN n),repeated E(ESNN n) times)
times)
times) times) layers of subscripts)
layers of subscripts)(E(E(ESNN n),repeated E(ESNN n,repeated E(ESNN
n,repreated ESNN n times))
times) times)),
repeated that many times)))}) that many times)} raised to itself E(E_{(that many times)},repeated E(E_{(E(that many
times))}) times) times.

Now that we have defined the Seriously Nominated Numbers...

The Transitional Nominee

What I temporarily consider as the largest named number with a prescribed method of calculation as the never-ending search for more qualifying numbers proceeds is not even an Epstein Number,but an Alphabet Number.

The one expressed by the

E_{((every integer starting with 1,repeated E(E(n^n... in a
power tower of nⁿ layers) layers of subscripts)(n^n...
in a power tower of nⁿ layers)) times...to the
last repetition of E_{(E_{(Epstein's First Seriously Nominated
Number layers of subscripts)(Epstein's First Seriously Nominated Number,
repeated E(Epstein's First Seriously Nominated Number) times)})}
)}
Seriously Nominated Number of

Z's,
all bracketed and individually separated by like number of colons in each instance.

Getting Really Serious

It now becomes time to again clean up the function and take it forward.

E_{(E(E(...(E(Transitional Nominee,repeated E(Transitional Nominee)
times) layers of subscripts)...(E(1),E(1,1),etc until we reach
E(every integer starting with 1,repeated E(E(n^n... in a
power tower of n^n... in a power tower of
E(nⁿ,repeated E(Transitional Nominee) times) layers)

layers of subscripts)(n^n...
in a power tower of E(nⁿ) layers)) times...to the
last repetition of E_{(E_{(E(Transitional Nominee)
layers of subscripts)(E(Transitional Nominee),
repeated E(Transitional Nominee,repeated E(Transitional Nominee)
times)})})}
is Epstein's First Really Seriously Nominated Number,or E¹R₁SNN.

To increment E¹R_nSNN to E¹R_n+1SNN one must substitute
(E_{((E¹R₁SNN)layers of subscripts)
(E(every integer starting with 1,repeated
E¹R₁SNN raised to itself in a power
tower of E¹R₁SNN layers,times,to the
last repetition of
E_{(E_{(E_{(E¹R₁SNN))}layers of
subscripts)
(E(E¹R₁SNN),
repeated E((E¹R₁SNN),repeated
E(E(E¹R₁SNN)times))})})} for every occurrence of "Transitional Nominee" in the above formula if n=1,and this formula,with every occurrence of E¹R_n-1SNN replaced by E¹R_nSNN, in every amended formula thereafter.

When n of E¹R_nSNN reaches E_{(E¹R_{E(E¹R₁SNN layers of
subscripts)((E¹R₁SNN,repeated
E(E¹R₂SNN) times))}SNN)} we have reached Epstein's First Really,Really Seriously Nominated Number or E²R₁SNN.

Hereafter,EⁿR_{E(E(EⁿR_E(n)SNN)
layers of subscripts)((EⁿR_E(n)SNN,repeated
E(EⁿR_E(n,n)SNN) times))}SNN = Eⁿ⁺¹R₁SNN.

The formula for incrementing the subscripted R-number in succeeding generations of the superscripted-prefixed R number is modified by requiring that the substitution of the expanded formula for each occurrence of the subscripted number take place not once,but EⁿR₁SNN times.

Of course n in ⁿR is like the number of an Epstein number just shorthand for (n,0,0,0,0...).
Thus,
E^{E(E(E(E(E(...E(n)
layers...(EⁿR₁SNN))))))}R₁SNN
is also E^(n,1)R₁SNN.
E^{E(E(E(E(E(...E(n,1)
layers...(E^(n,1)R₁SNN)))))}R₁SNN
is also E^(n,2)R₁SNN.
E^{E(E(E(E(E(...E(E(n),E(n)layers)
layers...(E(E^(n,n)R₁SNN),
E(E^(n,n)R₁SNN))))))))}R₁SNN
is also E^(n,1,1)R₁SNN.
E^{E(E(E(E(E(...E(E(E(n),E(n),E(n)),(E(n),E(n),E(n) layers)layers)
layers...(E(E^(n,n,n)R₁SNN),
(E^(n,n,n)R₁SNN),(E^(n,n,n)R₁SNN)))))))))}R₁SNN
is also E^(n,1,1,1)R₁SNN.
etc.

Armed with this escalation formula,we proceed to define the

First Kingmaker Number

As in,not taking the nomination itself,but has a role in defining it. As with the Transitional Nominee,it's an Alphabet Number,expressed by Z's of a quantity expressed by an Epstein Number.

The preceding superscript of R has as its comma-separated generations every integer starting with 1,repeated E(E^{(every integer from 1 to
E^{(E⁹R₉SNN)}R₉SNN)}R₉SNN) raised to itself in a power tower of E^{(every integer from 1 to E^E⁹R₉SNNR₉SNN) layers,times,to the
last repetition of
E_{(E_{(E_{(E^(9,9)R_E(9)SNN))}layers of
subscripts)
(E(E⁹R₉SNN),
repeated E((E⁹R₉SNN),repeated
E(E(E⁹R₉SNN)times))times)})})

The subscript is E^(9,9,9)R_E(9,9,9)SNN.

And the First Kingmaker Number is that many Zs
individually separated by as many colons within a set of brackets.}

Awarding the Next Transitional Nomination....

The superscript of the number that at this point bears my nomination as largest named number is
1.Every integer from 1 to E^{(First Kingmaker Number^{Kingmaker
Number} repetitions of every number from 1 to E(First Kingmaker
Number))}R_{(E(First Kingmaker Number))}SNN, each number repeated (E(n) layers of subscripts)(E(E(n) repeated E(n) times)).We shall call the designated maximum number here the Terminal Number.
2.Every possible sequence of any two integers,duplication permitted between 1 and the Terminal Number...each sequence ordered in ascending order of Epstein Number value potential,each repeated (E(the sequence) layers of subscripts)(E(E(the sequence) repeated E(the sequence) times)).
Then every possible sequence of three integers between 1 and the Terminal Number,likewise.
Then every possible sequence of four integers between 1 and the Terminal Number,likewise...et cetera.
3.After the last repetition of the entire sequence of integers, repeat steps 1 and 2 working out of the set of integers between 2 and the Terminal Number,with the number of layers of subscripts, and of the repetitions of each E(number) or E(sequence) in defining the number of repetitions,raised to a power tower with the Kingmaker Number as the base,followed by E(n)(or E(sequence)) layers of E(n) (or E(sequence)) followed by a number of ascending layers of First Kingmaker Numbers defined by the value of the E-layers.

4.Repeat step 3 with the set of integers between 3 and the Terminal Number,then the set between 4 and the Terminal Number,etc.
5.Repeat steps 1-4 E(First Kingmaker Number) times,with the number of subscripts/repetitions never resetting but continuing to be exponentiated throughout.

The subscript of this number is E(First Kingmaker Number repetitions of Terminal Number).

Since I was implored by a reader who stayed up until 1 AM reading the previous nomination to keep going...

Getting Extraordinarily Serious

E^{E_{E(E(...E_{(E(last transitional nominee layers of
subscripts)(last
transitional nominee,repeated E(last transitional nominee) times)}
layers of subscripts)...(E(Alphabet Number represented by last
transitional nominee Zs,repeated E(E(that Alphabet Number,repeated
E(that Alphabet Number) times) times)}}R_{E(E_{(last transitional nominee,repeated E(E(last
transitional nominee,repeated E(last transitional nominee)
times)times)}}SNN
is Epstein's First Extraordinarily Seriously Nominated Number,or EE₁SNN.

E_{E(E...E(EE_nSNN,repeated E(EE_nSNN,repeated...
(E(EE_nSNN,repeated E(EE_nSNN)times repetitions of
this)...times)times) layers of subscripts)
E(EE_nSNN,repeated E (EE_nSNN,repeated E(EE_nSNN,
repeated... (E(EE_nSNN,repeated E(EE_nSNN)times repetitions of
this)...times)times) layers of subscripts...(E(E(E(EE_nSNN)))
times)} = EE_n+1SNN.

When n reaches E(the following sequence:)
1.Every integer from 1 to E(E(EE₁SNN^{EE₂SNN...(power tower rising to
EE_{EE₉₉₉SNN}SNN)}) repetitions of every number from 1 to E(E(EE_{{REALLY::BIG::ALPHABET::NUMBER}}SNN,repeated E(EE_{{AN::EVEN::BIGGER::ALPHABET::NUMBER}}SNN times) ))SNN, each number repeated (E(EE_nSNN) layers of subscripts) (E(E(EE_nSNN) repeated E((EE_nSNN) times)). We shall call the designated maximum number here the first terminal number of this specification.
2.Every possible sequence of any two integers,duplication permitted between 1 and the terminal number...each sequence ordered in ascending order of Epstein Number value potential,each repeated (E(the sequence) layers of subscripts)(E(E(the sequence) repeated E(the sequence) times)).
Then every possible sequence of three integers between 1 and the terminal number,likewise.(always no limit on duplicate numbers)
Then every possible sequence of four integers between 1 and the terminal number,likewise...et cetera.
3.After the last repetition of the entire sequence of integers, repeat steps 1 and 2 working out of the set of integers between 2 and the terminal number,with the number of layers of subscripts, and of the repetitions of each E(number) or E(sequence) in defining the number of repetitions,raised to a power tower with the terminal number as the base,followed by E(E(n))(or E(E)sequence))) layers of E(E(n)) (or E(E(sequence))) followed by a number of ascending layers of EE_{n counting up from 1}SNN numbers defined by the value of the E-layers.
4.Repeat step 3 with the set of integers between 3 and the terminal number,then the set between 4 and the terminal number,etc.
5.Repeat steps 1-4 E(first terminal number) times,with the number of subscripts/repetitions never resetting but continuing to be exponentiated throughout.
6.Repeat steps 1-5 E(E...(E(first terminal number) layers of subscripts) (E(first terminal number),repeated E(first terminal number) times) times, substituting the number of repetitions used for the final sequence of numbers in the last cycle of step 5 for the first terminal number.

we have reached Epstein's First Extra Extraordinarily Seriously Nominated Number,or EE₁ESNN,which starts a new generation.

Incrementing EE_nESNN uses the formula used for incrementing EE_nSNN except that EE_nESNN is substituted for EE_nSNN on every occasion and in the specifications of the numbers of layers of subscripts and of the repetitions,this substitution of the full number of subscripts and repetitions for the specified number is then repeated EE_nESNN times.

EE_{EE_{EE_{...EE_nESNN layers...
EE_nESNN}ESNN}ESNN}ESNN =EⁿE₁ESNN.

When this latest n reaches E(the following sequence:)
1.Every integer from 1 to E(E(EE₁ESNN^{EE₂ESNN...(power tower rising to
EE_{EE_{(999 layers of subscripts)
EE₉₉₉ESNN}ESNN}ESNN)}) repetitions of every number from 1 to E(E(EE_{EE_{...E(999) layers of
subscripts...E_E(999)ESNN}ESNN,repeated
E(EE_{E_{E({AN::EVEN::LARGER::ALPHABET::NUMBER}) layers of
subscripts...E_E(99999)ESNN}ESNN times)
)})}ESNN, each number repeated (E(EE_nESNN) layers of subscripts) (E(E(EE_nESNN) repeated E((EE_nESNN) times)). We again call the designated maximum number here the first terminal number of this specification.
2.Every possible sequence of any two integers,duplication permitted between 1 and the terminal number...each sequence ordered in ascending order of Epstein Number value potential,each repeated (E( EE_{(the sequence)}ESNN) layers of subscripts)(E(E(EE_{(the sequence)}ESNN) repeated E(EE_{(the sequence)}ESNN times)).
Then every possible sequence of three integers between 1 and the terminal number,likewise.(always no limit on duplicate numbers)
Then every possible sequence of four integers between 1 and the terminal number,likewise...et cetera.
3.After the last repetition of the entire sequence of integers, repeat steps 1 and 2 working out of the set of integers between 2 and the terminal number,with the number of layers of subscripts, and of the repetitions of each E(number) or E(sequence) in defining the number of repetitions,raised to a power tower with the terminal number as the base,followed by E(E(EE_nESNN)) (or E(E(EE_sequenceESNN))) layers of E(E(EE_nESNN)) (or E(E(EE_sequenceESNN))) followed by a number of ascending layers of EE_{n counting up from 1}ESNN numbers defined by the value of the E-layers.
4.Repeat step 3 with the set of integers between 3 and the terminal number,then the set between 4 and the terminal number,etc.
5.Repeat steps 1-4 E(first terminal number) times,with the number of subscripts/repetitions never resetting but continuing to be exponentiated throughout.
6.Repeat steps 1-5 E(E...(E(EE_{first terminal number}ESNN) layers of subscripts)(E(EE_{first terminal number}ESNN),repeated (EE_{(EE_{first terminal number}ESNN)}ESNN layers of subscripts)(EE_{first terminal number}ESNN,repeated (EE_{first terminal number}ESNN)times)) times) times, substituting the number of repetitions used for the final sequence of numbers in the last cycle of step 5 for the first terminal number.

we reach another arbitary turning point number.
When we take the Alphabet Number represented by this many Zs each separated by a like number of colons and a layer of nesting curly and square brackets,we plug it into this:

E^{E^{E...(repeat this E^{...(repeat this E^{this
Alphabet Number}E_{(E(this Alphabet Number)}ESNN times)
(E^{(E(this Alphabet Number))}E_{E(This Alphabet
Number)}ESNN}
E_{(E(this Alphabet Number)}ESNN times)}
E_{(E(this Alphabet Number)}ESNN times)} E_{(E(this Alphabet Number)}ESNN

(The number of superscripting layers is invoked upon the number of superscripting layers the like number of times) to yield Epstein's First Very Extraordinarily Seriously Nominated Number,or EV₁ESNN,

Incrementing EV_nESNN uses the formula used for incrementing EE_nESNN except that EV_nESNN is substituted for EE_nESNN on every occasion and in the specifications of the numbers of layers of subscripts and of the repetitions,this substitution of the full number of subscripts and repetitions for the specified number is then repeated EV_{(EV_nESNN layers of
subscripts)(E(EV_nESNN),repeated E(EV_nESNN)
times)}ESNN times.

When the number of subscripts in EV_nESNN and this latest n reaches EV_{(the following
sequence:)}ESNN
1.Every integer from 1 to E(E(EV₁ESNN^{EV₂ESNN...(power tower rising to
EV_{EV_{(E(9,9,9) layers of subscripts)
EE_E(9,9,9)ESNN}ESNN}ESNN)}) repetitions of every number from 1 to E(E(EV_{EV_{...E(9,9,9) layers of
subscripts...EV_E(9,9,9)ESNN}ESNN,repeated
E(EV_{E_{E({AN::ENORMOUSLY::LARGER::ALPHABET::NUMBER}) layers of
subscripts...EV_E(9,9,9,9,9)ESNN}ESNN times)
)})}ESNN, each number repeated (E(EV_nESNN) layers of subscripts) (E(E(EV_nESNN) repeated E((EV_nESNN) times))times. We again call the designated maximum number here the "first terminal number" of this specification.
2.Every possible sequence of any two integers,duplication permitted between 1 and the terminal number...each sequence ordered in ascending order of Epstein Number value potential,each repeated (E( EV_{(the sequence)}ESNN) layers of subscripts)(E(E(EV_{(the sequence)}ESNN) repeated E(EV_{(the sequence)}ESNN times)).
Then every possible sequence of three integers between 1 and the terminal number,likewise.(always no limit on duplicate numbers)
Then every possible sequence of four integers between 1 and the terminal number,likewise...et cetera.
3.After the last repetition of the entire sequence of integers, repeat steps 1 and 2 working out of the set of integers between 2 and the terminal number,with the number of layers of subscripts, and of the repetitions of each E(number) or E(sequence) in defining the number of repetitions,raised to a power tower with the terminal number as the base,followed by E(E(EV_nESNN)) (or E(E(EV_sequenceESNN))) layers of E(E(EV_nESNN)) (or E(E(EV_sequenceESNN))) followed by a number of ascending layers of EV_{EV_{n counting up from 1}ESNN}ESNN numbers defined by the value of the E-layers.
4.Repeat step 3 with the set of integers between 3 and the terminal number,then the set between 4 and the terminal number,etc.
5.Repeat steps 1-4 EV_{(first terminal number)}ESNN times,with the number of subscripts/repetitions never resetting but continuing to be exponentiated throughout.
6.Repeat steps 1-5 E(E...(E(EV_{first terminal number}ESNN) layers of subscripts)(E(EV_{first terminal number}ESNN),repeated (E(EV_{first terminal number}ESNN)layers of subscripts) E(EV_{first terminal number}ESNN,repeated (EV_{E(first terminal number,repeated itself ultrexed
itself times to itself times)}ESNN)times)) times) times, substituting the number of repetitions used for the final sequence of numbers in the last cycle of step 5 for the first terminal number.
7.Repeat steps 1-6 EV_{(number of subscripts equal to the number of times
steps 1-5 were repeated in the previous repetition of step 6)(EV_{that number,
repeated that many times}ESNN,repeated itself cubed times)}ESNN times,substituting the number of repetitions used for the final sequence of numbers in the last cycle of step 6 for the first terminal number.
Steps 8- E(E...(E(EV_{first terminal number}ESNN) layers of subscripts)(E(EV_{first terminal number}ESNN),repeated (E(EV_{first terminal number}ESNN)layers of subscripts) E(EV_{first terminal number}ESNN,repeated (EV_{E(first terminal number,repeated itself ultrexed
itself times to itself times)}ESNN)times)) times) times). These steps follow the pattern laid out in Step 7,relating to Step (Step number -1).

we reach another passing Kingmaker-Number...
The Alphabet Number expressed by that many Zs each separated by colons,the first two by that many,each working rightward by the previous number ultrexed that number of times to itself is another, while if one uses that Alphabet Number as the first terminal number of the specification used to generate the previous Kingmaker-Number and then repeats this process that whole Kingmaker Number of times, one reaches Epstein's first Very,Very Extraordinarily Seriously Nominated Number,or E²V₁ESNN.

Incrementing E²V_nESNN uses the formula used for incrementing EV_nESNN except that E²V_nESNN is substituted for EV_nESNN on every occasion and in the specifications of the numbers of layers of subscripts and of the repetitions,this substitution of the full number of subscripts and repetitions for the specified number is then repeated EV_{(E²V_nESNN layers of
subscripts)(E(E²V_nESNN),repeated
E(E²V_nESNN,repeated
E²V_(E(n,n,n))ESNN times)
times)}ESNN times.

E²V_{E²V_{(E(2,3,4,...E²V₃ESNN,
each number repeated E²V_{that number}ESNN times)
layers of subscripts)
(E(2,3,4,...E²V₃ESNN,
each number repeated E²V_{that number popbled
itself times}ESNN times)}ESNN}ESNN

will serve as another passing Kingmaker-Number...
The Alphabet Number expressed by that many Zs each separated by colons,the first two by that many,each working rightward by raising the number of colons to the previous number ultrexed that number of times to (itself popbled itself-Zs ultrexed (itself-{Z:Z}s popbled itself-{Z::Z::Z}s popbled...(that-Kingmaker-Number-popbled of Zs layers of nesting))
was my standard-bearing nominee as highest named calculable number when a disk crash took this page offline in April 2017.

Getting this page back up calls for

The Resurrection Celebration Number

So let's ultrex the Alphabet Number expressed by E(every number from 2 to the previous nominee repeated itself ultrexed itself times to E(E_{(every number from 1 to itself cubed,repeated E_{(every number from 1 to itself to the 4th power,repeated...(so on until
E_{(every number from 1 to itself to the
E²V_E(itself)ESNN power)} is
reached)} layers of subscripts)}times) Z's E_{(WE,ARE,BACK,ONLINE,HALLELUJAH!)} times to its E_{(RESURRECTION,CELEBRATION)} power and popble that itself-ultrexed-twice-to-its-cube times.

Resurrection Celebration Number 2

Ultrex the Alphabet Number expressed by E(every number from 2 to the Resurrection Celebration Number repeated itself popbled itself times to E(E_{(every number from 1 to itself cubed,
repeated E_{(every number from 1 to itself to the 4th power,repeated...
(so on until E_{(every number from 1 to itself to the
E²V_E(itself)ESNN power)} is
reached)} layers of subscripts)}times) Z's each separated by that-many-raised-to-the-popbled-ordinal-of-the-Z colons E_{(LET,US,ALL,ROW,THE,BOAT,ASHORE,HALLELUJAH!)} times to its EV_{(RESURRECTION,CELEBRATION,QUOTH,THE,RAVEN,FOREVERMORE!)}ESNN power and popble that itself-popbled-ten-times-itself-popbled times.

The 2020 Vision Number

The creation of the Hyper-Operating Ultrex Function motivates a yet greater creation.

Popble the Alphabet Number with "Resurrection Celebration Number 2" Zs 2020 times. My next nominee as largest calculatable number is
(the result of that popbling)hu',_{E_(20,20)hu'}' .

The Blindingly Big Number and the Big LHOT

Of course no sooner had I announced the Hyper-Operating Ultrex Function than I realised the Simple Hyper-Operating Transformer Function would be enormously more powerful. Enough to make the 2020 Vision Number invisible.

So my next nominee as largest calculatable number was the Blindingly Big Number:
(2020 Vision Number)(s)',_{(2020 Vision number popbled E(20,20)
times)s'}'

The day after that I realized that function was weak tea and came up with the Ladder Hyper-Operating Transformer Function which could enable substantially greater numbers,that would blind one to the Blindingly Big Number for good reason... So my NEW nominee as largest calculatable number was the Big LHOT:
(Blindingly Big Number)(l)',_{(Blindingly Big Number popbled
E(T,W,E,N,T,Y,T,W,E,N,T,Y) times)l'}'

The 2021 Contender

Having created the Super Conway function I decided to give a place in the sun to

(Big LHOT popbled Big LHOT times)_{s}→

The Restless Number

Later in 2021 I decided to transfer the spotlight to The Restless Number,
The Alphabet Number with 2021 Contender Zs popbled E_{(AND,{HEEEEEERE:S},THE,2021,CONTENDER,[The 2021 Contender])} times.

The 2022 Contender and new generations

To celebrate the new quetta- and ronna- prefixes and the new URL here (lekno.ws replacing put.com) we embark on our most complex calculation yet.

Popble the Restless Number E_{(TWENTY,TWENTY,TWO)} quettillia-ronnillia'quettillio-ronnillio-quettillion times and call the resulting number Second Order a.

Place Second Order a in the Alphabet Number function, performing the function that many times,to yield Second Order A,and continue through every calculation on this page to redefine new-order Alphabet Numbers,Epstein Numbers,and so forth, using the new order versions in the definitions.

When you reach the Second Order Restless Number,popble it as many times as the first was,and repeat all the calculations in defining new Third Order Alphabet Numbers,Epstein Numbers,etc. used to define their counterparts.

Repeat all steps before this one twice.

Repeat all steps before this one three times.

Repeat all steps before this one four times...

...and so forth until you have made the repetition cycle of producing new order numbers that is Restless-Number-popbled-E_{(TWENTY,
TWENTY,TWO)}-quettillia-ronnillia'quettillio-ronnillio-quettillion-times steps long...producing the latest-order reincarnation of the Restless Number.

The 2022 Contender is the latest-order Alphabet Number with that many Zs each separated by that many colons.

The 2023 Contenders...

A new year,another vast calculation of an even larger number:

Perform the Alphabet Number function on the 2022 Contender E_{(T,W,O,T,H,O,U,S,A,N,D,A,N,D,T,W,E,N,T,Y,T,H,R,E,E)} times and designate this as the latest order a.

Perform all the calculations,using this order,to turn that a into the 2022 Contender of that new order.

Repeat all steps before this one twice.

Repeat all steps before this one three times.

Repeat all steps before this one four times...

...when you reach the 2022-Contender-steps-number of repetition cycles,"reverse gear with a square"... the next set of repetitions is 2022-Contender-minus-one squared,then 2022-Contender minus-two-squared,and so on down the squares until you get to a cycle of four repetitions,then do three-cubed repetitions,four-cubed, and so on until you get up to 2022-Contender-minus-two cubed, then again reverse incrementing the exponent until you get down to three to the fourth,reverse incrementing starting with four to the fifth,up to 2022-Contender-minus-three to the fifth,reverse using sixth powers,and so on with just-one-short cycles reversing with higher powers until there is no room to reverse. At that point,create a new-order a and do all the steps to reach that order's 2022 Contender, and repeat ALL foregoing steps 2022-Contender-popbled times. The 2023 Contender is the latest-order Alphabet Number with that many Zs each separated by 2023 times that many colons.

...oh,and the second 2023 Contender is the latest-order Alphabet Number with E(2023,Contender,[the 2023 Contender],... repeated 2023 Contender times,E(E((2023,Contender,[the 2023 Contender],... repeated 2023-Contender-squared times,E(E(E(2023,Contender,[the 2023 Contender],...repeated 2023-Contender-cubed times,...and so on until the cycle of 2023-Contender-raised-to-2023-Contender-popbled-2023-Contender-times times, followed by the same sequence with ContendeR substituted for each Contender,followed by the same sequence with ContendEr,and so on with each case combination in ascending order until CONTENDER is used) Zs each separated by E(2023) times that many colons.

The 2024 Contender...

What's a new year without a new contender? Treat the Second 2023 Contender as a and perform all operations needed to take original a to the Second 2023 Contender,then repeat this cycle, then take the number reached after the repetition as original a and perform all calculations necessary to take original a to the number reached after that repetition,repeat this cycle twice,then take the number reached after that cycle as original a and perform all calculations needed to take originala to that number,then repeat this cycle three times,take the then resulting number as original a...and so forth until you have reached the cycle repeated E_{(THE,2024,CONTENDER,IS,BIGGER,THAN,THE,SECOND,2023,CONTENDER,
BY,A,LONG,SHOT...repeated E_{(THE,SECOND,2023,CONTENDER)}
times...the foregoing repeated E₂₀₂₄ times)} times. The 2024 Contender is the Alphabet Number with E(E(E(E(E(...E(2024)the just-reached number Zs.
Prove another comparably calculable number is bigger,and I'll go further...
If this number feels too small,popble the number the-number-popbled times, and repeat until the feeling goes away...or
Keep counting...
Louis Epstein

First edition January 4,2016
Second Nominated Number added January 5,2016
HTML tweaks and Third Nominated Number added January 6,2016
Expansion to the Transitional Nominee on January 9th 2016,and new nominee,followed by Extraordinarily Serious extension, January 12th.Further nominees January 13th and 15th. Enhancement with ultrex & nominee tweaks February 4th-11th.
Correction & E²VESNN incrementation added April 4th.
New nominee tweaks July 8th 2016 and March 4th 2017.
Resurrection Celebration September 11th 2017, second celebration on the 18th,some fixes the 21st.

Revision introducing the 2020 Vision Number April 24,2020,the Blindingly Big Number April 25,2020,and the Big LHOT April 26,2020.
Link revised for renaming of SHOT April 27,2020.
Revision introducing 2021 Contender January 4,2021.
Restless Number introduced October 2,2021,with a typo fix.
2022 Contender introduced November 28,2022
2023 Contender introduced February 11,2023,typo fixes March 21.
Second 2023 Contender introduced November 12,2023.
2024 Contender introduced January 18,2024.
Do you just want a shortcut to the ultimate number? (not computable)
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