The language of non-extendable strings of a regular language is regular
The definition does not ask for a maximal length string in $L$ but it looks for the strings that cannot be extended into a longer string in the language: $x\in L$, but no $xy \in L$ for nonempty $y$.

This means that repetitions are no problem. For $L = a^*b$ in fact $\mathrm{Max}(L) = L$ as no string can be extended to a longer string in $L$. But for $L = ba^*$ we have $\mathrm{Max}(L) = \varnothing$,as every string can be extended.

If you start with a (deterministic!) FSA, you can easily see which strings can be extended, and which not. How?