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There are a number of lowerbounds in circuit complexity.

Some well-known examples:

1. Parity is not in $AC^0$:

2. $mod_p$ is not in $AC^0[q]$.

3. 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/