What's the “smallest” complexity class for which an $\omega \hspace{.02 in}(n)$ circuit lower bound is known? I believe the answers to this question give classes such that for all polynomials $p$,

there is a problem in the class which does not have circuits of size $p(n)$.

However, I'm asking about circuit size $\omega \hspace{.02 in}(n)$.

$\big(\hspace{-0.07 in}\left\langle \hspace{-0.04 in}0^{\hspace{.02 in}0}\hspace{-0.03 in},\hspace{-0.04 in}1^{\hspace{-0.03 in}1}\hspace{-0.03 in},2^{\hspace{.02 in}2}\hspace{-0.04 in},\hspace{-0.03 in}3^1\hspace{-0.04 in},\hspace{-0.03 in}4^4\hspace{-0.03 in},\hspace{-0.03 in}5^1\hspace{-0.04 in},\hspace{-0.03 in}6^{\hspace{.03 in}6}\hspace{-0.03 in},\hspace{-0.03 in}7^1\hspace{-0.03 in},\hspace{-0.03 in}8^8\hspace{-0.03 in},\hspace{-0.03 in}9^1\hspace{-0.03 in},...\hspace{-0.05 in}\right\rangle \:$ is super-linear but not $\omega \hspace{.02 in}(n)$.

Although such even-odd behavior could be handled by padding, one might instead

have extremely long streaks of super-polynomial values between low values.)
$S^p_2$ 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.