User blog:Unknown95387/Help please

This is totally noob question for you, but I need to ask you something.

I have an idea for a number called \(mag(3)\), where mag is a function. Mag is a function with \(n\) gag's, and \(gag(n)\) is equal to \(A(n,n)\) in Ackermann function (\(n\)'s instead of \(x\) and \(y\)). So there are 3 gag's to make up mag(3). The first \(gag(3)\) is 61. The second one is much much larger and I think it's equal to \(f_\omega(60)\) (Largest number I've ever got to, and we're not at the end !). Now there's the problem:

I've tried many ways to compute the third \(gag\) and I ended with \(f_{\omega}(60)\{f_{\omega}(60)\{f_{\omega}(60)\}f_{\omega}(60)\}f_{\omega}(60)\) or \(f_{\omega}(60)\{A(f_{\omega}(60))\}f_{\omega}(60)\) with usage of Ackermann numbers. So finally here are my 2 questions:

What is this number equal in BEAF or FGH ?

In which class of numbers is this ?