What's the “smallest” complexity class for which an $\omega \hspace{.02 in}(n)$ circuit lower bound is known?
$S^p_2$ and $PP$ are both known not to have $n^k$-circuits for any fixed k and there is no known containment between them. Details in my blog post. 

Update: As Rickey Demer points out, these results do not necessarily give a language with a lower bound for all $n$ in $S_2^p$. I think the $\Delta^p_3$ is probably the best known. Since $PP$ has complete sets you might be able to get a all $n$ bound but I don't have a full proof.