  We study the nonlinear response of a driven cavity QED system in the extreme
strong coupling regime where the saturation photon number is below one by many
orders of magnitude. In this regime, multi-photon resonances within the
Jaynes--Cummings spectrum up to high order can be resolved. We identify an
intensity and frequency range of the external coherent drive for which the
system exhibits bistability instead of resonant multi-photon transitions. The
cavity field evolves into a mixture of the vacuum and another quasi-classical
state well separated in phase space. The corresponding time evolution of the
outgoing intensity is a telegraph signal alternating between two attractors.
