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Most complex networked systems across biology and engineering are externally driven by perturbations, imposing non-equilibrium and sometimes non-stationary response dynamics. Strong external driving may induce tipping and cascading failure, yet state-of-the-art methods from network dynamics have focused on linear response theory suitable for weak driving signals only. Here we report nonlinear responses emerging generically in driven nonlinear dynamical systems yet are absent from most text book examples. Moreover, at some critical (large) driving amplitude, responses cease to stay close to a given operating point and may diverge – the system tips. As standard response theory fails to predict tipping amplitudes, even at arbitrarily high orders, we propose self-consistency conditions that capture the genuinely nonlinear response dynamics. Our novel approach uncovers a generic pondermomotive route to tipping. It may help to quantitatively predict intrinsically nonlinear response dynamics as well as bifurcations emerging at large driving amplitudes in non-autonomous dynamical systems. We highlight several application directions and a broad field of open theoretical and methodogical questions.