We report the observation of the higher order frequency shift due to the trapping field in a $^{87}$Sr optical lattice clock. We show that at the magic wavelength of the lattice, where the first order term cancels, the higher order shift will not constitute a limitation to the fractional accuracy of the clock at a level of $10^{-18}$. This result is achieved by operating the clock at very high trapping intensity up to $400 $kW/cm$^2$ and by a specific study of the effect of the two two-photon transitions near the magic wavelength.