Using two-photon Bragg spectroscopy, we study the energy of particle-like excitations in a strongly interacting homogeneous Bose-Einstein condensate, and observe dramatic deviations from Bogoliubov theory. In particular, at large scattering length $a$ the shift of the excitation resonance from the free-particle energy changes sign from positive to negative. For an excitation with wavenumber $q$, this sign change occurs at $a\approx 4/(\pi q)$, in agreement with the Feynman energy relation and the static structure factor expressed in terms of the two-body contact. For $a\gtrsim 3/q$ we also see a breakdown of this theory, and better agreement with calculations based on the Wilson operator product expansion. Neither theory explains our observations across all interaction regimes, inviting further theoretical efforts.