  We consider a sub-wavelength periodic layered medium whose slabs are filled
by arbitrary linear metamaterials and standard nonlinear Kerr media and we show
that the homogenized medium behaves as a Kerr medium whose parameters can
assume values not available in standard materials. Exploiting such a parameter
availability, we focus on the situation where the linear relative dielectric
permittivity is very small thus allowing the observation of the extreme
nonlinear regime where the nonlinear polarization is comparable with or even
greater than the linear part of the overall dielectric response. The behavior
of the electromagnetic field in the extreme nonlinear regime is very peculiar
and characterized by novel features as, for example, the transverse power flow
reversing. In order to probe the novel regime, we consider a class of fields
(transverse magnetic nonlinear guided waves) admitting full analytical
description and we show that these waves are allowed to propagate even in media
with $\epsilon<0$ and $\mu >0$ since the nonlinear polarization produces a
positive overall effective permittivity. The considered nonlinear waves
exhibit, in addition to the mentioned features, a number of interesting
properties like hyper-focusing induced by the phase difference between the
field components.
