User blog comment:Chronolegends/X Hierarchy/@comment-28606698-20171023134608

We can invent a lot of interesting variations on theme of the fast-growing hierarchy.

For example we can define for natural number $$n>0$$ and ordinal number $$\alpha\geq 0$$ where $$\alpha[n]$$ denotes the n-th element of the fundamental sequence assigned to the limit ordinal $$\alpha$$. If $$\alpha>0$$ then $$f_{\alpha}(n)$$ always is a power of 10,
 * $$f_0(n)=10\cdot n$$
 * $$f_{\alpha}^0(n)=n$$
 * $$f_{\alpha}^{m+1}(n)=f_\alpha(f_{\alpha}^m(n))$$
 * $$f_{\alpha+1}(n)=f_{\alpha}^n(10)$$
 * $$f_{\alpha}(n)=f_{\alpha[n]}(10)$$ iff $$\alpha$$ is a countable limit ordinal

or at placing ordinal directly as subscript of a natural number


 * $$n_0=10\cdot n$$
 * $$n_{\alpha,1}=n$$
 * $$n_{\alpha,m+1}=(n_{\alpha,m})_{\alpha}$$
 * $$n_{\alpha+1}=10_{\alpha,n}$$
 * $$n_{\alpha}=10_{\alpha[n]}$$ iff $$\alpha$$ is a countable limit ordinal

and a lot of other variations