User blog:IAmNotATRex/My Extension to the Fast Growing Hierarchy

I don't know much about googology or ordinal numbers, but I thought it would be fun to try to create my own extension to the fast growing hierarchy. I don't know if this has been done before or how this compares to other notations or functions, so I'd appreciate it if someone could compare this to other forms of notation or other functions.

I don't have much experience formatting with LaTeX, so please forgive me if something doesn't look good.

Definition
I haven't really thought of a good name or notation, so I'll just use temporary notation for this blog post and refer to this as "my extension."

My new extension uses the following format: $$Fn_a(b)$$, where $$F$$ is the name of the function.

For $$n=0$$, $$F$$ simply reduces to the fast growing hierarchy: $$F0_{a}(b)=f_{a}(b)$$.

For $$n>0$$ and $$a=0$$, $$Fn_{0}(b)=Fn-1_{Fn-1_{\cdots{Fn-1_{0}(b)}\cdots}(b)}(b)$$, where $$Fn-1$$ is repeated $$b$$ number of times.

For $$a>0$$, $$Fn_{a}(b)=Fn-1_{Fn-1_{\cdots{Fn-1_{a}(b)}\cdots}(b)}(b)$$, where $$Fn-1$$ is repeated $$Fn_{a-1}^{b}(b)$$ number of times.

Examples
Starting out using the number $$3$$ in an example, $$ F1_{0}(3)=F0_{F0_{F0_{0}(3)}(3)}(3) \\ =f_{f_{f_{0}(3)}(3)}(3) \\ =f_{f_{4}(3)}(3)$$

$$F1_{1}(3)$$ would be the same, except there would be $$F1_{0}^{3}(3)$$ number of layers and the subscript at the bottom would be $$1$$ instead of $$0$$.

Increasing $$n$$ from $$1$$ to $$2$$ just increases all of the numbers after $$f$$ by $$1$$.

For example, $$ F2_{0}(3)=F1_{F1_{F1_{0}(3)}(3)}(3)$$ and $$F2_{1}(3)$$ would have $$F2_{0}^{3}(3)$$ number of layers.

Hopefully you were able to understand my first blog post. I'm not sure if my extension is useful at all, but I did create it just for fun.