As I understand it, you can't meaningfully ask about the number of layers in the human brain.
Layers are an abstraction we use to make artificial neural networks dramatically faster to work with using linear algebra. By keeping each node interacting only with the layers above and below, we make the computations a lot nicer for our model of computation.
In the brain, neurons link freely to other arbitrary neurons based on adaption processes we (or at least I) don't really understand.
The brain also lacks a clear idea of direction to count the layers along, since it has innumerable different inputs coming in at all times, and the resulting signals interact all over the place.
The most meaningful analog would probably be to ask "How many neuron firings typically occur between an external stimulus and a response to that stimulus?" Even that is extremely rough though, because through evolution a lot of 'short circuit' structures have formed in our bodies. The gag reflex is obviously triggered by sensory input, but it probably doesn't check with your frontal lobe before firing the appropriate muscles.