Atomically thin rhenium disulphide (ReS2) is a member of the transition metal dichalcogenide family of materials. This two-dimensional semiconductor is characterized by weak interlayer coupling and a distorted 1T structure, which leads to anisotropy in electrical and optical properties. Here we report on the electrical transport study of mono- and multilayer ReS2 with polymer electrolyte gating. We find that the conductivity of monolayer ReS2 is completely suppressed at high carrier densities, an unusual feature unique to monolayers, making ReS2 the first example of such a material. Using dual-gated devices, we can distinguish the gate-induced doping from the electrostatic disorder induced by the polymer electrolyte itself. Theoretical calculations and a transport model indicate that the observed conductivity suppression can be explained by a combination of a narrow conduction band and Anderson localization due to electrolyte-induced disorder.