What kind of problems require more than 2 hidden layers? I've read that the most of the problems can be solved with 1-2 hidden layers.

How do you know you need more than 2? For what kind of problems you would need them (give me an example)?
Formally, a single hidden layer is sufficient to approximate a continuous function to any desired degree of accuracy, so in that sense, you never need more than 1. This is called the Universal Approximation Theorem.

Finding the best topology for a given problem is an open research problem. As far as I know, there are few universal 'rules of thumb' for this.

For a given problem, one option is to apply a neuroevolutionary approach such as NEAT, which attempts to find a topology that works well for the problem at hand.