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
ℏq 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.