User blog comment:IAmNotATRex/My Extension to the Fast Growing Hierarchy/@comment-35870936-20180820052650/@comment-35870936-20180821043644

Hmmm, I guess I missed that it has \(Fn_{a-1}^b(b)\) layers. That would explain why I thought it was so weak.

Notice that \(F0_{F0_{...F0_1(x)...}(x)}(x)\) with n layers, has the growth rate of \(f_{\omega+1}(x)\). This isn't hard to see, since \(F0_n(x)\) is equal to \(f_n(x)\).

Let's test F1_1.

We have \(F1_1(x) = F0_{F0_{...F0_1(x)...}(x)}(x)\), with \(F1_0^x(x) = f_{\omega+1}^x(x) = f_{\omega+2}\) layers. So F1_1(x) has a growth rate of about \(f_{\omega+2}(x)\). F1_2(x) will have a growth rate of \(f_{\omega+3}(x)\), and in general, F1_n(x) will have a growth rate of \(f_{\omega+n+1}(x)\). F2_0(x) will have a power of (f_{\omega2+1}(x)\), F2_n(x) will have a power of (f_{\omega2+n+1}(x)\), and in general, Fa_n(x) will have a power of (f_{\omega*a+n+1}(x)\). So your notation has a limit of \(\omega^2\) in the FGH.

It is important to note that nearly all the power from this notation comes from the \(Fn_{a-1}^b(b)\) expression in your blog post. The \(Fn_a(b)=Fn-1_{Fn-1_{\cdots{Fn-1_a(b)}\cdots}(b)}(b)\) expression doesn't really help at all in making the notation more powerful.