Whereas the Kerr nonlinearity is well understood in the perturbative limit of nonlinear optics, there is considerable discussion about its functional form and magnitude at extreme intensities, at which point matter starts to ionize. Here, we introduce a concept to answer this question and theoretically analyze its feasibility. We demonstrate that seeded Kerr instability amplification provides clear signatures from which functional form and magnitude of the Kerr nonlinearity can be extracted in the non-perturbative limit of nonlinear optics.