Recently, Ruddlesden-Popper 2D perovskite (RPP) solar cells and light-emitting diodes (LEDs) have shown promising efficiencies and improved stability in comparison to 3D halide perovskites. Here, the exciton recombination dynamics is investigated at room temperature in pure-phase RPP crystals (C6H5C2H4NH3)2(CH3NH3)n-1PbnI3n+1 (n = 1, 2, 3, and 4) by time-resolved photoluminescence (TRPL) in a large range of power excitations. As the number of perovskite layers increases, we detect the presence of an increasing fraction of out-of-equilibrium free carriers just after photoexcitation, on a picosecond time scale, while the dynamics is characterized by the recombination of excitons with long lifetime spanning several tens of nanoseconds. At low excitation power, the TRPL decays are nonexponential because of defect-assisted recombination. At high fluence, defects are filled and many-body interactions become important. Similar to other 2D systems, exciton-exciton annihilation (EEA) is then the dominant recombination path in a high-density regime below the Mott transition.