= Solution
Represent a <function> by its graph of <Kuratowski ordered pairs>. For $f:\omega\to\omega^2$,
$$
\operatorname{rk}(f)=\sup_{n<\omega}\big(\max(n,f(n))+3\big)\leq\omega^2.
$$
The bound is attained by $f(n)=\omega n$, which is a <cofinal function> into $\omega^2$. Taking the rank of the set of all such functions adds one at this attained maximum. Thus $\boxed{\operatorname{rk}((\omega^2)^\omega)=\omega^2+1}$. The extra successor is essential: unlike the previous product, this set actually contains a member of rank $\omega^2$.
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