  The interplay of strong disorder and superconductivity is a topic of
long-term interest in condensed matter physics. Here we explore the nonlinear
THz response of superconducting NbN films close to the 3D metal-insulator
transition. For the least disordered samples, the magnitude of the nonlinear
$\chi^{(3)}$ response follows the temperature dependence of the superfluid
density as expected. In contrast, for high disorder samples near the
metal-insulator transition the $\chi^{(3)}$ nonlinearity persists to
temperatures as high as even 4 times the $T_c$ of the cleanest sample. We
discuss the possible origins of this remarkably large nonlinearity, including
the possibility that it arises in an enhancement of the temperature scales of
superconductivity close to localization. Our work highlights the importance of
finite frequency nonlinear THz experiments in detecting superconducting
correlations even into regions where long-range ordered superconductivity does
not persist.
