Highly nonlinear optical materials with strong effective photon-photon interactions are required for ultrafast and quantum optical signal processing circuitry. Here we report strong Kerr-like nonlinearities by employing efficient optical transitions of charged excitons (trions) observed in semiconducting transition metal dichalcogenides (TMDCs). By hybridising trions in monolayer MoSe at low electron densities with a microcavity mode, we realise trion-polaritons exhibiting significant energy shifts at small photon fluxes due to phase space filling. We find the ratio of trion- to neutral exciton-polariton interaction strength is in the range from 10 to 100 in TMDC materials and that trion-polariton nonlinearity is comparable to that in other polariton systems. The results are in good agreement with a theory accounting for the composite nature of excitons and trions and deviation of their statistics from that of ideal bosons and fermions. Our findings open a way to scalable quantum optics applications with TMDCs.