Let $T:Σ^*\to Σ^*$ be an operation such that $T(L)$ is regular for all regular languages $L \in Σ^*$.
Is it possible to prove $T^∞(L)$ is regular?
Colleague Apass Jack already warned about the dangers of infinity, and he also indicated a very simple example that shows that a very simple iteration leads to a non-regular language. Case closed, but I like to add an observation: iterating simple local substitutions lead to Turing power, not just non-regularity.
The single steps of a Turing machine van be encoded, and performed with a regular operation. The instruction "on reading $a$ in state $q$, write $b$, move left and change to state $p$" is coded as rule $aq \mapsto pb$ and is extended to longer strings containing a single state as $\alpha aq \beta \mapsto \alpha pb\beta $. This operation can be extended to sets of instructions, and will map a regular language into a regular language. However,iterating them will actually generated Turing machine computations!
(I have an answer somewhere with this observation, but cannot find it.)