Hi Yaro,
I do not believe there is any drawback in using multilayer board, excepting manufacturability issues, which are translated into prices and mechanical properties of the PCB. It is also harder to troubleshoot.
The more layers you have it is easier to route appropriately all the signals and guaranteed that you have solid planes for both ground and power. Also, the thinner the dielectric more the signal and return path are close together. This minimizes inductance, therefore minimizing cross-talk.
As far as I know there is none cheap prototype service exceeding 4 layers. My advice is contacting or experimenting with online calculators of several manufacturers. For example in Europe, Multicircuits, Eurocircuits, Wurth Elektronik, PCB-POOL and try to find the cheapest possible technology selection that fulfils your requirements.
Alex