$\mathsf{NC^1}$ circuit evaluation
As far as I know, $\mathsf{NC^1}$ evaluation is not known to be in $\mathsf{NC^1}$, and is conjectured to be outside $\mathsf{NC^1}$. See 


Emil Jerabek, "On theories of bounded arithmetic for $\mathsf{NC^1}$",  Ann. Pure Appl. Logic 2011