Effects of high-Mach numbers and high initial amplitudes on the evolution of the single-mode Richtmyer-Meshkov shock-wave induced hydrodynamic instability are studied using theoretical models, experiments, and numerical simulations. Two regimes in which there is a significant deviation from the linear dependence of the initial velocity on the initial perturbation amplitude are defined and characterized. In one, the observed reduction of the initial velocity is primarily due to large initial amplitudes. This effect is accurately modeled by a vorticity deposition model, quantifying both the effect of the initial perturbation amplitude and the exact shape of the interface. In the other, the reduction is dominated by the proximity of the shock wave to the interface. This effect is modeled by a modified incompressible model where the shock wave is mimicked by a moving bounding wall. These results are supplemented with high initial amplitude Mach 1.2 shock-tube experiments, enabling separation of the two effects. It is shown that in most of the previous experiments, the observed reduction is predominantly due to the effect of high initial amplitudes.