[maze-le]

> And would such a theory have any practical use?

Yes, sureley. In Computablilty Theory you have the famous Ackermann-function[1]. It is actually an operator-extension, like you just described. It is important, because it grows overexponentially, but is still computable (unlike e.g. the busy-beaver-function).

[1]: https://en.wikipedia.org/wiki/Ackermann_function

[infogulch]

Given that Graham's Number is an (over-)extension of the Ackermann-function that means that it's still computable, correct?

Given the series leading up to Graham's Number, G = g_64 (g_1, g_2, ...), BB(n) will outgrow g_n, right? If so what's the smallest n such that BB(n) > g_n?

[infogulch]

After a bit of research I found that there is a proof [0][1] that BB(12) > g_1. But that's still a ways from g_2, let alone g_64.

[0]: "A Lower Bound on Rado's Sigma Function for Binary Turing Machines" by Milton Green (1964)

[1]: https://en.wikipedia.org/wiki/Busy_beaver#Known_values_for_....