A "simple" language outside $CFL \cup coCFL$? I am looking for a language L with the following properties:


L should not be context-free.
L's complement should not be context-free. (Everything you see in textbooks as prime examples of non-context-free languages seem to fail this second requirement.)
L shouldn't be too hard, For example, I know that undecidable languages fit the first two requirements, but what I want is a simpler language which can be recognized by a slightly "improved" automaton model, e.g. a probabilistic pushdown automaton.
How about $L:=\{a^{n^2}\mid n\in\mathbb{N}\}$? It is easy to see that $L$ and its complement are not regular, and hence (as we are dealing with a unary alphabet) not context-free.