  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
multi-exciton correlations. Here, by means of two-dimensional coherent
spectroscopy, we examine excitonic many-body effects in pure, single-layer
(PEA)$_2$PbI$_4$ (PEA = phenylethylammonium). We determine the binding energy
of biexcitons --- correlated two-electron, two-hole quasiparticles --- to be
$44 \pm 5$\,meV at room temperature. The extraordinarily high values are
similar to those reported in other strongly excitonic two-dimensional materials
such as transition-metal dichalchogenides. Importantly, we show that this
binding energy increases by $\sim25$\% upon cooling to 5\,K. Our work
highlights the importance of multi-exciton correlations in this class of
technologically promising, solution-processable materials, in spite of the
strong effects of lattice fluctuations and dynamic disorder.
