This paper deals with the homogenization of a sequence of non-linear conductivity energies in a bounded open set Omega of R-d, for d >= 3. The energy density is of the same order as a(epsilon)(x/epsilon)vertical bar Du(x)vertical bar(p), where epsilon -> 0, a(epsilon) is periodic, a is a vector-valued function in W-1,W-p(Omega; R-m) and p > 1. The conductivity a, is equal to 1 in the "hard" phases composed by N >= 2 two by two disjoint-closure periodic sets while a, tends uniformly to 0 in the "soft" phases composed by periodic thin layers which separate the hard phases. We prove that the limit energy, according to Gamma-convergence, is a multi-phase functional equal to the sum of the homogenized energies (of order 1) induced by the hard phases plus an interaction energy (of order 0) due to the soft phases. The number of limit phases is less than or equal to N and is obtained by evaluating the Gamma-limit of the rescaled energy of density epsilon(-p) a(epsilon)(y)vertical bar Dv(y)vertical bar(p) in the torus. Therefore, the homogenization result is achieved by a double Gamma-convergence procedure since the cell problem depends on a.