Is there any language for which $\overline{L^*}=\big(\overline L\big)^*$?
No, thanks for Ariel's correction, since for every $L$ it holds that $\epsilon \not\in \overline{(L^*)}$  and $\epsilon \in (\overline{L})^*$ and therefore $ \overline{(L^*)}\neq (\overline{L})^*$