There are a number of lowerbounds in circuit complexity.
Some well-known examples:
Parity is not in $AC^0$:
$mod_p$ is not in $AC^0[q]$.
k-Clique is not in $AC^0$.
There are also results for monotone circuits and for a few nesting of majority/threshold gates.
You can find references and more result in the second part of
S. Arora and B. Barak, "Computational Complexity: A Modern Approach", http://www.cs.princeton.edu/theory/complexity/