User blog comment:Mh314159/A hopefully powerful new system/@comment-39585023-20190630030857/@comment-35470197-20190630153047

I might be wrong in the analysis, but it is true that FGH is so powerful that it is very difficult to imagine how powerful it is as long as we have not learned it.

For example, \(f_{\omega}\) is very strong, \(f_{\omega + 1}\) is incomparably stronger than \(f_{\omega}\), \(f_{\omega + 2}\) is insanely stronger than \(f_{\omega + 1}\). Adding \(1\) to the ordinal in FGH gives a great jump of the resulting function. If we have a recursion approximately \(f_{\omega^2}\), repeating similar recursions just yields \(f_{\omega^2+\omega}\) or something like that.

Anyway, I am certain that you can soon go beyond \(\omega^{\omega^{\omega}}\) if you learn FGH, because you have sufficient ability to write down complicated rule sets.