  Strong inter-particle interactions between polaritons have traditionally
stemmed from their exciton component. In this work, we impart a strong photonic
nonlinearity to a polaritonic mode by embedding a nonlinear polymethine dye
within a high-Q all-metal microcavity. We demonstrate nonlinear microcavities
operating in the ultrastrong coupling regime with a normalized coupling ratio
of 62\%, the highest reported to date. When pumping the lower polariton branch,
we observe tunable third-harmonic generation spanning the entire visible
spectrum, with internal conversion enhancements more than three orders of
magnitude larger than in bare films. Transfer matrix calculations indicate that
the observed enhancements are consistent with the enhanced pump electric field.
