Organic-inorganic lead halide perovskites have emerged in recent years as semiconductor materials for various optoelectronic and photovoltaic applications. The exciton binding energy is an important parameter because the formation of excitons can potentially hamper charge separation in solar cells. However, it remains a challenge to experimentally determine the exciton binding energy in hybrid bromine-based perovskites: the obtained values have a tens meV spread. Here we present in detail the near-band edge photoluminescence study in CH3NH3PbBr3 single crystals under different photoluminescence excitation densities with a 405 nm laser diode. We show that by using high laser excitation intensities it is possible to make a direct measurement of the exciton binding energy, which we find to be only 12 meV at low temperatures, lower than has been previously determined. In the low temperature orthorhombic phase, besides the free exciton emission at 2.25 eV, we observed the broad yellow-light emission at 2.16 eV, which exhibited a red shift with the increasing temperature and a blue shift with increasing laser excitation intensity. Based on the excitation power density and temperature dependence of the photoluminescence spectra, we interpreted yellow-light emission as a recombination of bound excitons and a donor-acceptor pair transition. The results obtained are essential for a better understanding of the electronic properties of these materials and provide a guideline for their further applications with improved performance.