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Nonlinear optomechanical coupling is the basis for many potential future experiments in quantum optomechanics (e.g., quantum nondemolition measurements, preparation of nonclassical states), which to date have been difficult to realize due to small nonlinearity in typical optomechanical devices. Here we introduce an optomechanical system combining strong nonlinear optomechanical coupling, low mass, and large optical mode spacing. This nanoscale “paddle nanocavity” supports mechanical resonances with hundreds of femtograms of mass that couple nonlinearly to optical modes with a quadratic optomechanical coupling coefficient g(2)>2π×400 MHz/nm2, and a single-photon to two-phonon optomechanical coupling rate of Δω0>2π×16 Hz. This coupling relies on strong phonon–photon interactions in a structure whose optical mode spectrum is highly nondegenerate. Nonlinear optomechanical readout of thermally driven motion in these devices should be observable for T>50 mK, and measurement of phonon shot noise is achievable. This shows that strong nonlinear effects can be realized without relying on coupling between nearly degenerate optical modes, thus avoiding the parasitic linear coupling present in two-mode systems.