Can a non-regular language $L$ have a non regular $L^*$?

I have been looking around and i cant seem to find an example of such case that a non-regular $L$ has a non regular $L^*$. Is it possible? If so, can you provide an example of such case please?

Take the Dyck language $D$. It is context-free but not regular and satisfies $D^* = D$.

StackExchange Computer Science Q#140480, answer score: 4