User blog:MilkyWay90/another attempt at making a fast-growing function

\(A(n) = n + 1\) which growth rate is \(f_0(n)\)

\(A(a, b) = \underbrace{A(A(\ldots A(b) \ldots))}_{a}\) which growth rate is \(f_{0}^{a}(b)\) or \(f_{1}(b)\)

\(A(a, b, c) = A(A(A(\ldots A(a, b) \ldots, \ldots A(a, b) \ldots), A(\ldots A(a, b) \ldots, \ldots A(a, b) \ldots)), A(A(\ldots A(a, b) \ldots, \ldots A(a, b) \ldots), A(\ldots A(a, b) \ldots, \ldots A(a, b) \ldots)))\) Not really sure about the growth rate.

\(A(a, b, B, c) = \underbrace{A(A(A(\ldots A(a, b, B) \ldots)}_c, A(\ldots A(a, b, B) \ldots), B, A(\ldots A(a, b, B) \ldots)), A(A(\ldots A(a, b, B) \ldots), A(\ldots A(a, b, B) \ldots), B, A(\ldots A(a, b, B) \ldots)), B, A(A(\ldots A(a, b, B) \ldots), A(\ldots A(a, b, B) \ldots), B, A(\ldots A(a, b, B) \ldots)))\)

\(B(a) = \underbrace{A(a, a, \ldots, a)}_a\)

\(B(a, b, C, c) = \underbrace{B(B(B(\ldots A(a, b, C) \ldots)}_c, B(\ldots B(a, b, C) \ldots), C, B(\ldots B(a, b, C) \ldots)), B(B(\ldots B(a, b, C) \ldots), B(\ldots B(a, b, C) \ldots), C, B(\ldots B(a, b, C) \ldots)), C, B(B(\ldots A(a, b, C) \ldots), B(\ldots B(a, b, C) \ldots), C, A(\ldots B(a, b, C) \ldots)))\)

And so on...

Put all of the above functions in one set named set

\(set = {A, B, \ldots, Z, AA, AB, \ldots, ZZ, AAA, \ldots}\)

define \(A_{0}(a) = set_{a}(\underbrace{a, a, \ldots, a}_{a})\)

\(A_{n}(a) = set_{a}(\underbrace{a, a, \ldots, a}_{a})\) where the starting \(A\) is \(A_{n - 1}\)

What is the growth rate of \(A_{n}(a)\) and how can I improve the growth rate? Please leave the answer or any comments below.