Middle-growing hierarchy

The middle-growing hierarchy is a hierarchy created by Googology Wiki user.

The rules are as following:
 * \(m(0,n)=n+1\)
 * \(m(\alpha+1,n)=m(\alpha,m(\alpha,n))\)
 * \(m(\alpha,n)=m(\alpha[n],n)\)

Up to \(\omega^\omega\)
\begin{eqnarray*} m(0,n) &=& n + 1 \\ m(1,n) &=& n + 2 \\ m(2,n) &=& n + 4 \\ m(3,n)  &=& n + 8 \\ m(k,n) &=& n + 2^k \\ m(\omega,n) &=& n + 2^n \\ m(\omega+1,n) &=& n + 2^n + 2^{n + 2^n} \\ m(\omega+2,n) &=& n + 2^n + 2^{n + 2^n} + 2^{n + 2^n + 2^{n + 2^n}} > 2^{2^{2^n}} \\ m(\omega+m,n) &>& 2^En\#(m+1) > 2\uparrow\uparrow(m+1) \\ m(\omega2,n) &>& 2\uparrow\uparrow(n+1) \\ m(\omega3,n) &>& 2\uparrow\uparrow\uparrow(2^n) \\ m(\omega m,n) &>&  2\uparrow^m(2^n) \\ m(\omega^2,n)  &>& 2\uparrow^n(2^n) \\

m(\omega^2+\omega,n)  &>& \lbrace n,2^n,1,2 \rbrace \\ m(\omega^22,n)  &>& \lbrace n,2^n,n,2 \rbrace \\ f_{\omega^3}(n) &>&  \lbrace n,2^n,n,n \rbrace \\ f_{\omega^m}(n) &>& \lbrace n,m+1 (1) 2 \rbrace \\ f_{\omega^{\omega}}(n) &>& \lbrace n,n+1 (1) 2 \rbrace > \lbrace n,n (1) 2 \rbrace \\ \end{eqnarray*}

We see that the middle-growing hierarchy catches the fast-growing hierarchy at \(\omega^{\omega}\), and generally, it does so at all multiples of it.