Proton transfer processes play an important role in many fields of chemistry. In dilute basic aqueous solutions, proton transfer from water molecules to hydroxide ions is aided by "presolvation", i.e., thermal fluctuations that modify the hydrogen-bonding environment around the proton-receiving OH(-) ion to become more similar to that of a neutral H2O molecule. In particular at high concentrations, however, the underlying mechanisms and especially the role of the counterions are little understood. As a prototypical case, we investigate aqueous NaOH solutions using molecular dynamics simulations employing a reactive high-dimensional neural-network potential constructed from density functional theory reference data. We find that with increasing concentration the predominant proton transfer mechanism changes from being "acceptor-driven", i.e., governed by the presolvation of OH(-), to "donor-driven", i.e., governed by the presolvation of H2O, and back to acceptor-driven near the room-temperature solubility limit of 19 mol/L, which corresponds to an extremely solvent-deficient system containing only about one H2O molecule per ion. Specifically, we identify concentration ranges where the proton transfer rate is mostly affected by OH(-) losing an accepted hydrogen bond, OH(-) forming a donated hydrogen bond, H2O forming an accepted hydrogen bond, or H2O losing a coordinated Na(+). Presolvation also manifests itself in the shortening of the Na(+)-OH2 distances, in that the Na(+) "pushes" one of the H2O protons away.