  We dress atoms with multiple-radiofrequency fields and investigate the
spectrum of transitions driven by an additional probe field. A complete
theoretical description of this rich spectrum is presented, in which we find
allowed transitions and determine their amplitudes using the resolvent
formalism. Experimentally, we observe transitions up to sixth order in the
probe field using radiofrequency spectroscopy of Bose-Einstein condensates
trapped in single- and multiple-radiofrequency-dressed potentials. We find
excellent agreement between theory and experiment, including the prediction and
verification of previously unobserved transitions, even in the
single-radiofrequency case.
