User blog comment:Mh314159/Alpha numbers (and beyond)/@comment-35470197-20191007022858/@comment-39585023-20191015102435

Thank you,  but I think you underestimate the f{n,n,n...}(x) function. Here is my analysis of a few expressions with two small index terms that go far beyond Graham's number, which I understand has growth rate w+1. The number growth very fast with a small increase in the final index term, and much faster still with more index terms. I will put up some examples of f{1,2} and beyond soon if you are interested.

f‹1,1›(1) = f‹m,0›m(m) = f‹m›m(m) where m = f‹1,1›(0) = f‹0,1›(2) = f‹f‹1,0›(2),0›(2) = f‹f‹1›(2)›(2) = f‹8›(2) so f‹1,1›(1) iterates the f‹f‹8›(2)› function f‹8›(2) times. f‹8›(2) is at least equal to 2^^^^^^^^^(5) and it specifies the number up up arrows in f‹1,1›(1) which puts f‹1,1›(1) between G2 and G3 in the Graham-Gardner sequence. The functional power m is inconsequential for functions with a single large index of the form f‹p›.

f‹1,1›(2) = f‹m,0›m(m) = f‹m›m(m) where m = f‹1,1›(1) > G3

f‹1,1›(63) = f‹m›m(m) with m = f‹1,1›(62) is greater than the Graham-Gardner number

f‹1,1›2(1) = f‹1,1›(f‹1,1›(1)) and if f‹1,1›(1) = p then f‹1,1›(p) = f‹m,0›m(m) = f‹m›m(m) where m = f‹1,1›(p-1) and m recurses p times before reaching f‹1,1›(0) = f‹0,1›(2) = f‹8›(2). This is equivalent to iterating the Graham-Gardner sequence more than G2 times, or G(G2).

f‹2,1›(1) = f‹m›m(m) where m = f‹2,1›(0) = f‹1,1›(f‹3›(3)) > G(2^^^^5)

f‹2,1›(2) = f‹m›m(m) where m = f‹2,1›(1) so this iterates Graham more than G(2^^^^5) times.