New propagation regimes for light arise from the ability to tune the dielectric permittivity to extremely low values. Here, we demonstrate a universal approach based on the low linear permittivity values attained in the ε-near-zero (ENZ) regime for enhancing the nonlinear refractive index, which enables remarkable light-induced changes of the material properties. Experiments performed on Al-doped ZnO (AZO) thin films show a sixfold increase of the Kerr nonlinear refractive index (n_{2}) at the ENZ wavelength, located in the 1300 nm region. This in turn leads to ultrafast light-induced refractive index changes of the order of unity, thus representing a new paradigm for nonlinear optics.