What's the "smallest" complexity class for which a superlinear circuit bound is known? Apologies for asking a question that must surely be in a lot of standard references.  I'm curious about exactly the question in the title, in particular I am thinking of Boolean circuits, no depth bound.  I put "smallest" in quotes to allow for the possibility there are multiple different classes, not known to include each other, for which a superlinear 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 $k$.