Asteroseismic studies of red giants generally assume that the oscillation modes can be treated as linear perturbations to the background star. However, Kepler observations show that the oscillation amplitudes increase dramatically as stars ascend the red giant branch. Thus, the linear approximation may not always be valid. Here we analyze the stability of mixed modes in red giants to nonlinear, resonant three-wave interactions. We find that modes near the peak of the observed power spectrum are unstable to three-wave interactions over a broad range of stellar mass and evolutionary state. We construct large networks of nonlinearly coupled modes consisting of stochastically driven parent modes coupled to resonant secondary modes (daughters, granddaughters, etc.). By solving the amplitude equations that describe the mode dynamics, we determine the nonlinear saturation energy of the modes in our networks. We find that for stars whose frequency of maximum power numax < 100 microHz, the parent amplitudes are five to ten times smaller, on average, than in the linear calculations. For stars with 150 < numax < 200 microHz, the average amplitudes are one to two times smaller. We discuss these results in the context of various asteroseismic observables, including the population of red giants that exhibit dipole modes with unusually small amplitudes.