Forum:Feeding FGH into FGH

I'm a newbie in googology, and I have some questions about FGH and ordinals. FGH looks like this: $$f_\alpha(n)$$ while $$\alpha$$ can be an integer or an ordinal. what I want to do is to feed it into it self: $$f_{f_\alpha(\omega)}(n)$$ but here's some problem about this:

1.how to define $$f_\alpha(\omega)$$  when   $$\alpha$$  is an limit ordinal? fundamental sequence is totally useless.

2.what's the limit of this? is that the fix point of $$f_\alpha(\omega)=\alpha$$? if so, the ordinal $$\alpha$$ is probably $$\Gamma_0$$ I guess,which is not even as powerful as the original FGH!!!

3.$$\Gamma_0$$ can be expressed as $${\{\omega,\omega,1,2}\}$$ in BEAF. it seems like that it is possible to range $$\Gamma_0$$ by using $$f_\alpha(\omega)$$. what's the problem between 2 and 3?