User blog comment:MilkyWay90/Yet another attempt at making a fast-growing function/@comment-35470197-20180709135900/@comment-35470197-20180712230751

The updated version contains many ambiguous explanations. For example, to what extent you allow recursions. I guess that you are just considering primitive recursions. The way to count symbols is not so precise.

At least, all of the ambiguity could be removed easily by using Goedel numbers, sets of functions and transformations, or formal languages of second order arithmetic. Therefore I do not care about the problem here.

The most important problem is that the function is an uncomputable one, and hence even if it is well-defined, it does not yield computable large numbers.

If you prefer computable large numbers, then you need to realise the definition more precisely using computing process. Otherwise, it is ok. (I note that if one uses uncomputable functions, then the growth rate can easily become larger than \(\omega_1^{\textrm{CK}}\), which is greater than the growth rate of any computable ones.)