Polaritonic devices exploit the coherent coupling between excitonic and photonic degrees of freedom to perform highly nonlinear operations with low input powers. Most of the current results exploit excitons in epitaxially grown quantum wells and require low temperature operation, while viable alternatives have yet to be found at room temperature. Here we show that large single-crystal flakes of two-dimensional layered perovskite are able to sustain strong polariton nonlinearities at room temperature with no need to be embedded in an optical cavity. In particular, exciton-exciton interaction energies are measured to be remarkably similar to the ones known for inorganic quantum wells at cryogenic temperatures, and more than one order of magnitude larger than alternative room temperature polariton devices reported so far. Thanks to their easy fabrication, large dipolar oscillator strengths and strong nonlinearities, these materials hold great promises to realize actual polariton devices at room temperature.