  We study strongly nonlinear disordered media using a functional method. We
solve exactly the problem of a nonlinear impurity in a linear host and we
obtain a Bruggeman-like formula for the effective nonlinear susceptibility.
This formula reduces to the usual Bruggeman effective medium approximation in
the linear case and has the following features: (i) It reproduces the weak
contrast expansion to the second order and (ii) the effective medium exponent
near the percolation threshold are $s=1$, $t=1+\kappa$, where $\kappa$ is the
nonlinearity exponent. Finally, we give analytical expressions for previously
numerically calculated quantities.
