Electrophoresis is the motion of particles relative to a surrounding fluid driven by a uniform electric field. In conventional electrophoresis, the electrophoretic velocity grows linearly with the applied field. Nonlinear effects with a quadratic speed vs field dependence are gaining research interest since an alternating current field could drive them. Here, we report on the giant nonlinearity of electrophoresis in a nematic liquid crystal in which the speed grows with the fourth and sixth powers of the electric field. The mechanism is attributed to the shear thinning of the nematic environment induced by the moving colloid. The observed giant nonlinear effect dramatically enhances the efficiency of electrophoretic transport.