Exponential gap on neural network layers
The paper that people usually cite is Almost Optimal Lower Bounds for Small Depth Circuits, which appears in STOC 1986. The main result pertaining to your question is: 


  There exists a family of functions admitting linear size, depth $k$ circuits (of unbounded fan-in AND/OR and NOT) that requires exponential size at depth $k-1$. 


What's possible even more relevant is the fact that $\mathsf{TC}^0$ admits an exponential separation between depth 3 and depth 2. This is relevant because threshold gates are commonly used in deep networks.