I devised a notation that encodes every polynomial and goes further into hyper-exponential functions, and then applies functional operators onto these
The notation is defined as follows:
a[0,b]c = b, regardless of c
a[b,c]d = a[b,c-1][b-1,d]a, b >= 1
a[b,c][0,d]f = a[b,c]a + d
a[b,c][d,f]g = a[b,c][d,f-1]
For example,
2[2,2]2
= 2[2,1][1,2]2
= 2[2,1][1,1][0,2]2
= 2[2,1][1,1]2 + 2
= 2[2,1][0,2]2 + 2
= 2[2,1]2 + 4
= 2[1,2]2 + 4
= 2[1,1][0,2]2 + 4
= 2[1,1]2 + 6
= 2[0,2]2 + 6
= 8
If both arguments are the same, then [n,a][n-1,b][n-2,c]...[3,mn-2][2,mn-1][1,mn][0,mn+1] corresponds to axn + bxn-1 + cxn-2 + ... + mn-2x3 + mn-1x2 + mnx + mn+1. This means that n[3,1]n is n3, n[4,1]n is n4, and so on. The second argument of the bracketed expressions is extensible into negative and real-valued arguments.
This may not seem too impressive, but a bracketed expression can have more than 2 arguments. n[1,1,1]m is equal to n[m,1]n, which is just nm. Higher expressions are defined like so:
a[1,1,b]c = a[1,1,b-1][1,1,1]a = b*aa
a[1,1[1],1]b = a[1,1,b]a
a[1,1[b],1]c = a[1,1[b-1],c]a
a[1,2,1]b = a[1,1[b],1]a = aa+b
a[b,1,1]c = a[b-1,c,1], b >= 2
a[b,c,d]f = a[b,c,d-1][b,c-1[f-1],d], c >= 2
a[b,c,1]d = a[b,c-1[d],1]a
a[b,c[d],1]f = a[b,c[d-1],f]a
a[b,c[d],f]g = a[b,c[d]f-1]a + a[b,c[d-1],a]a
For example,
3[1,1,1]3
= 3[3,1]3
= 3[2,2][1,2][0,3]3
= 3[2,2][1,2]3 + 3
= 3[2,2][1,1]3 + 6
= 3[2,2]3 + 9
= 3[2,1][1,2]3 + 12
= 3[2,1][1,1]3 + 15
= 3[2,1]3 + 18
= 3[1,2]3 + 21
= 3[1,1]3 + 24
= 3 + 24
= 27
This is just a relatively low-level example, and doesn't really showcase what 3-argument bracketed expressions can do.
4[4,4,4]4
= 4[4,4,3][4,4,1]4
= 4[4,4,3][4,3[4],1]4
= 4[4,4,3][4,3[3],4]4
= 4[4,4,3][4,3[3],3]4 + 4[4,3[2],4]4
= 4[4,4,3][4,3[3],3]4 + 4[4,3[2],3]4 + 4[4,3[1],3]4 + 4[4,3,4]4
= 4[4,4,3][4,3[3],3]4 + 4[4,3[2],3]4 + 4[4,3[1],3]4 + 4[4,3,3][4,3,1]4
= 4*44*4^4 = 41025 = 22050 = 129268024285244029202859506754679807841776410678861936128521381710098620555471563572788805646091653854754871843687592077976478236601963684380352609545793132482523509469203984367000791001558608427184230553536270273107168874570479024647352377353904681882326583408145220171550303566164263234430209596495721542087657333558673369682739899146258277979424704141305288232311222637324104770833841256601034371456708466903774414873847429430393404609206583617614790452933148924908502738843280838900628406907725293878714170322626791740183989073411994552862100665557749342174408541732918418581273914419808592774223414444238384922624
The real fun begins when we apply functional operators to the encoded expression. A functional power recursively applies the function to a certain value. For instance, for a function F, F3(n) = F(F(F(n))). To apply a functional operator to an expression, simply add the functional operator after the bracketed expressions:
n[...][...]......[...](Fm) = n[...][...]......[...]n applied m times
For example,
2[7,4][4,6][3,10][2,7][1,15](F8) = (n -> 4n7 + 6n4 + 10n3 + 7n2 + 15n)8(2) = (n -> 4n7 + 6n4 + 10n3 + 7n2 + 15n)7(746) = (n -> 4n7 + 6n4 + 10n3 + 7n2 + 15n)6(514318072559117816162)
I defined two googolisms based on functional powers so far. The first, the powsigmagol, is equal to 10[10,1][9,1][8,1][7,1][6,1][5,1][4,1][3,1][2,1][1,1][0,1](F100), or n10 + n9 + n8 + n7 + n6 + n5 + n4 + n3 + n2 + n + 1 applied 100 times to 10. This number is "slightly" larger than googolplex, and has exactly 10045757490560240830928234202706579516985551580649045014557748187973571343755159726235759057152594226 digits, the first 900 and last 200 of which are given below.
66561072175997161203646950106362705954889518579072880362567781769286752003802006345461406424815121079033237206814232116925159659838929845823765579808419776213275392320854060894624395829076996451344929334844809653693609500291115841196859433805452892949991001673460503422773724685736930928005402892844484997504926906954561373638740761786678234792710913703627597452908292893416872196302922585400764409153542181992919248474929541733250676095064255530593984934516987038133176760213107486880708226409322248345650126477656306455239987064209428797446978995896431586481775559592585245828591274768438056200710006397951501843573681157817757409487348249360866505746101151974760788247965817895626357468802124430693108538487513697306538437757879911264468018777293629372054140711933204645831217668860017373222842099488247672280043817548531455606121185839092059792558731210195527618108830948303713543044212372021980…………………………………………………………………………………………………...…………………………………………………………………………………………………………………………………………………………………….63317731414196173747311945039742803245116854565164876117842310297686327561336980960692214086645470023312957228311530702207674118560580942634748375272634104344033676527536103561865018498081901253146061