Let $m^{m^m} = x$ and $n^{n^n} = y$. WLOG assume that $m\leq n$ and $x\leq y$. Rearrange the inequality to get $n^y-n^x\geq m^y-m^x$. This is equivalent to $n^x(n^{y-x}-1)\geq m^x(m^{y-x}-1)$. Since $n^x\geq m^x\geq 0$ and $n^{y-x}\geq m^{y-x}\geq 0$, the inequality always holds and we are done.