Nonlinear Compton scattering has been observed in the collision of a low-emittance 46.6-GeV electron beam with terawatt pulses from a Nd:glass laser at 1054 and 527 nm wavelengths in an experiment at the Final Focus Test Beam at SLAC. Peak laser intensities of ${10}^{18}\phantom{\rule{0ex}{0ex}}\mathrm{W}/{\mathrm{cm}}^{2}$ have been achieved, corresponding to a value of 0.6 for the parameter $\ensuremath{\eta}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}e{E}_{\mathrm{rms}}/m{\ensuremath{\omega}}_{0}c$. Results are presented for multiphoton Compton scattering in which up to four laser photons interact with an electron, in agreement with theoretical calculations.