Engineering strong interactions between optical photons is a challenge for quantum science. Polaritonics, which is based on the strong coupling of photons to atomic or electronic excitations in an optical resonator, has emerged as a promising approach to address this challenge, paving the way for applications such as photonic gates for quantum information processing1 and photonic quantum materials for the investigation of strongly correlated driven-dissipative systems2,3. Recent experiments have demonstrated the onset of quantum correlations in exciton-polariton systems4,5, showing that strong polariton blockade6-the prevention of resonant injection of additional polaritons in a well delimited region by the presence of a single polariton-could be achieved if interactions were an order of magnitude stronger. Here we report time-resolved four-wave-mixing experiments on a two-dimensional electron system embedded in an optical cavity7, demonstrating that polariton-polariton interactions are strongly enhanced when the electrons are initially in the fractional quantum Hall regime. Our experiments indicate that, in addition to strong correlations in the electronic ground state, exciton-electron interactions leading to the formation of polaron-polaritons8-11 have a key role in enhancing the nonlinear optical response of the system. Our findings could facilitate the realization of strongly interacting photonic systems, and suggest that nonlinear optical measurements could provide information about fractional quantum Hall states that is not accessible through their linear optical response.