With strongly bound and stable excitons at room temperature, single-layer, two-dimensional organic-inorganic hybrid perovskites are viable semiconductors for light-emitting quantum optoelectronics applications. In such a technological context, it is imperative to comprehensively explore all the factors---chemical, electronic, and structural---that govern strong multiexciton correlations. Here, by means of two-dimensional coherent spectroscopy, we examine excitonic many-body effects in pure, single-layer (${\mathrm{PEA})}_{2}{\mathrm{PbI}}_{4}$ (PEA = phenylethylammonium). We determine the binding energy of biexcitons---correlated two-electron, two-hole quasiparticles---to be $44\ifmmode\pm\else\textpm\fi{}5$ meV at room temperature. The extraordinarily high values are similar to those reported in other strongly excitonic two-dimensional materials such as transition-metal dichalcogenides. Importantly, we show that this binding energy increases by $\ensuremath{\sim}25%$ upon cooling to 5 K. Our work highlights the importance of multiexciton correlations in this class of technologically promising, solution-processable materials, in spite of the strong effects of lattice fluctuations and dynamic disorder.