The quantum Rabi model provides the framework for describing a two-level system interacting with a strong oscillating field beyond the rotating-wave approximation. We experimentally map the dressed-state spectrum, revealing multiple higher-order dressed-state transitions predicted by the model. The experiment is realized using proton spins in flowing water in a Rabi-type setup with a strong off-resonant magnetic dressing field. The measured transition energies show quantitative agreement with theoretical predictions. These results establish precision nuclear-spin systems as a platform for exploring quantum Rabi physics in regimes beyond the rotating-wave approximation and open new possibilities relevant to nuclear magnetic resonance and related spin-based measurements.