We report experiments on high-amplitude sound wave propagation in an acoustic metamaterial composed of an air-filled waveguide periodically side-loaded by holes. In addition to the linear viscothermal and radiation losses, high amplitude sound waves at the locations of the sideholes introduce nonlinear losses. The latter result in an amplitude-dependent reflection, transmission, and absorption, which we experimentally characterize. First, we evidence that nonlinear losses change the nature of the device from a reflective to an absorbing one, showing thepossibility to use the system as a nonlinear absorber. Second, we study the second-harmonic generation and its beating phenomenon bothexperimentally and analytically. We find that when considering the propagation of both the fundamental and the second harmonic, nonlinear losses cannot be neglected. Our results reveal the role of nonlinear losses in the proposed device and also provide a quite accurate analytical model to capture the effect of such losses.