Size of complement of context-free language
From the proof that determining if a CFL ${L}$ = $\Sigma^*$ is undecidable, the set of strings $ID_0\#ID_1^R\#ID_2\#ID_3^R\#\ldots\#ID_t$ where $ID_0,ID_1,\ldots,ID_t$ is a list of the configurations of an accepting nondeterministic TM, is the complement of a context-free language. So $|\overline{L}_n|$ can basically be any computable function less than exponential.