Inelastic scattering processes typically introduce friction among carriers and reduce the transport properties of photons, phonons, and electrons. However, we predict that in contrast to the role in reducing thermal conductivity, four-phonon scattering dominates the near-field radiative heat transfer (NFRHT) in both boron arsenide (BAs) and boron antimonide. Including four-phonon scattering results in a nearly 400-fold increase in the total heat flux between two BAs thin films compared to three-phonon scattering alone. This nonintuitive enhancement arises from the large number of NFRHT channels activated by four-phonon scattering, outcompeting the effect of decreased coupling strength of surface phonon polaritons at the resonance frequency. Additionally, we point out that four-phonon scattering decreases NFRHT in certain other systems.