  We study the Compton scattering of light by free electrons inside a
hyperbolic medium. We demonstrate that the unconventional dispersion and local
density of states of the electromagnetic modes in such media can lead to a
giant Compton shift and dramatic enhancement of the scattering cross section.
We develop an universal approach for the study of coupled multi-photon
processes in nanostructured media and derive the spectral intensity function of
the scattered radiation for realistic metamaterial structures. We predict the
Compton shift of the order of a few meVs for the infrared spectrum that is at
least one order of magnitude larger than the Compton shift in any other system.
