A generalized regular expression is like a regular expression but with one more operation allowed: complementation. The (generalized) star-height of a generalized regular expression is the maximal number of nested Kleene stars. The star-height of a language is the minimal star-height of a regular expression describing it. It is not known if there is even a language of star-height 2.
So I imagine the langage $((aa)^*bb(aa)^*bb(aa)^*)^*(aa)^*$ of words consisting of concatenations of $aa$ and $bb$, with an even number of $bb$, is of generalized star-height $1$. But I couldn't prove it. In the paper Some results on the generalized star-height problem (Pin, Straubing, Thérien), lemma 6.1 (the transfer lemma) cannot be applied because both $(bb)^*$ and $(aa)^*$ are of star-height 1.
I saw elsewhere some languages which were conjectured to be of star-height 2 and they were more complex than the one I give, so it is very probably of star-height 1. Is it the case?