This article addresses the measurement of the nonlinear modes of highly flexible structures vibrating at extreme amplitude, using a Phase-Locked Loop experimental continuation technique. By separating the motion into its conservative and dissipative parts, it is theoretically proven for the first time that phase resonance testing organically allows for measurement of the conservative nonlinear modes of a structure, whatever be its damping law, linear or nonlinear. This result is experimentally validated by measuring the first three nonlinear modes of a cantilever beam. Extreme amplitudes of motion (of the order of 120° of cross section rotation for the first mode) are reached for the first time, in air with atmospheric pressure condition, responsible for a strong nonlinear damping due to aeroelastic drag. The experimental backbone curves are validated through comparison to the conservative backbone curves obtained by numerical computations, with an excellent agreement. The classical trends of cantilever beams are recovered: a hardening effect on the first nonlinear mode and softening on the other modes. The nonlinear mode shapes are also measured and compared to their theoretical counterparts using camera capture. Finally, it is shown that the damping law can be estimated as a by-product of the phase resonance measurement of the conservative nonlinear modes. As an original result, the damping law is observed to be highly nonlinear, with quadratic and cubic evolutions as a function of the structure’s amplitude.