Quantum fluctuations in low-dimensional systems and near quantum phase transitions have significant influences on material properties. Yet, it is difficult to experimentally gauge the strength and importance of quantum fluctuations. Here we provide a resonant inelastic x-ray scattering study of magnon excitations in Mott insulating cuprates. From the thin film of SrCuO₂, single- and bi-magnon dispersions are derived. Using an effective Heisenberg Hamiltonian generated from the Hubbard model, we show that the single-magnon dispersion is only described satisfactorily when including significant quantum corrections stemming from magnon-magnon interactions. Comparative results on La₂CuO₄ indicate that quantum fluctuations are much stronger in SrCuO₂ suggesting closer proximity to a magnetic quantum critical point. Monte Carlo calculations reveal that other magnetic orders may compete with the antiferromagnetic Néel order as the ground state. Our results indicate that SrCuO₂—due to strong quantum fluctuations—is a unique starting point for the exploration of novel magnetic ground states.