Light propagating through random media produces characteristic speckle patterns, directly related to the large multitude of scattering events. These complex dynamics remarkably display robustness to perturbation of the incoming light parameters, maintaining correlation in the scattered wavefront. This behavior is known as the optical memory effect. Here we unveil the properties of the nonlinear optical memory effect, which occurs when an optothermal nonlinearity perturbs the random material. The effect is characterized through a series of pump and probe experiments in silica aerogel, in the visible range. This additional degree of freedom further generalizes the memory effect, opening the road to applications based on the nonlinear response of random media.