The response of a bistable system to a low-frequency driving force is demonstrated to display a giant nonlinearity: a sinusoidal force can induce a nearly rectangular signal in the system even when its amplitude is small. The nonlinearity comes about through the force modulating the effective activation energies of fluctuational transitions between the stable states, which in turn can give rise to strong modulation of the transition probabilities and the populations of the states for low noise intensities. Theoretical results are shown to be in good agreement with the results of analog and digital simulations.