  Moir\'e excitons promise a new platform with which to generate and manipulate
hybrid quantum phases of light and matter in unprecedented regimes of
interaction strength. We explore the properties in this regime, through studies
of a Bose-Hubbard model of excitons coupled to cavity photons. We show that the
steady states exhibit a rich phase diagram with pronounced bi-stabilities
governed by multi-photon resonances reflecting the strong inter-exciton
interactions. In the presence of an incoherent pumping of excitons we find that
the system can realise one- and multi-photon lasers.
