Hierarchy theorems for circuit depth
A paper of Klawe, Paul, Pippenger, and Yannakakis gives an hierarchy theorem for constant depth monotone formulas:
http://dl.acm.org/citation.cfm?id=808717

Specifically, for every $k$ it gives a function that can be computed by a formula of depth $k$ and size $n$ but requires formulas of depth $k-1$ of size $\exp(n^{1/k})$.