Turing Machine that computes maximum steps of halting machines
Hint: If you could compute $f(m,n)$, then given a Turing machine $M$ with $m$ states and an input $x$ of length $n$, the Turing machine will either stop within $f(m,n)$ steps or never halt.

This argument actually shows that it is impossible to compute any upper bound on $f(m,n)$. In other words, $f(m,n)$ grows faster than any computable function.