We demonstrate giant nonlinearity of the kinetic inductance Lₖ in a superconducting NbN/Al hybrid strip. In contrast to ordinary superconductors, where the maximum observed current-induced variation of Lₖ does not exceed tens of percent, we observe a 50-fold change in Lₖ. Physically, this effect is connected with the proximity-induced superconductivity in the Al layer and the specific choice of parameters for which the dependence of the superconducting current I on the Cooper pair momentum ℏ q exhibits a plateau at a current I^* it well below the depairing current. Formally, this leads to Lₖ ∼ dq/dI → ∞ as I → I^*. We believe that our results open a new route for creating the next generation of highly sensitive detectors of electromagnetic radiation, calorimeters, parametric amplifiers, and magnetometers.