We study numerically and experimentally the highly nonlinear dynamical regime (far above the primary instability threshold) of a one-dimensional spatially extended feedback system. The spatiotemporal dynamics is very complex and is characterized by the emission of a spatial frequency supercontinuum accompanied by the appearance of abnormally intense localized peaks in transverse patterns. We perform statistical analysis of this highly nonlinear regime in terms of the probability density function (PDF) of the peak intensities rather than usual tools such as correlation functions. We find that the statistics of these peak intensities is described very well by the generalized gamma (GG) probability density function and determine its three parameters which can be used as quantitative indicators of the transition from the weakly to the highly nonlinear regime. Most interestingly, we discover that in the highly nonlinear regime the GG PDF converges to the gamma probability density function with the shape parameter equal to 3/2. This limit corresponds to the Rayleigh probability density function of the peak amplitudes for the oceanic waves. This behavior of the PDF can be an indicator of the universality of the highly nonlinear regime for other processes involving supercontinua and chaos.