What's the "smallest" complexity class for which a superlinear circuit bound is known?
I believe that the smallest such classes known are $S_2P$ (Cai, 2001), $PP$ (Vinodchandran, 2005), and $(MA \cap coMA)/1$ (Santhanam, 2007).  All of these are indeed known to not be in $SIZE(n^k)$ for each constant $k$.