We study the scattering of polaritons by free electrons in hyperbolic photonic media and demonstrate that the unconventional dispersion and high local density of states of electromagnetic modes in composite media with hyperbolic dispersion can lead to a giant Compton-like shift and dramatic enhancement of the scattering cross section. We develop a universal approach to study multiphoton processes in nanostructured media and derive the intensity spectrum of the scattered radiation for realistic metamaterial structures.