We study four- and higher order wave mixing of continuous coherent waves on a single superconducting artificial atom. Narrow side peaks of different orders of nonlinearity resulting from elastic multiphoton scattering on the atom are observed and investigated. We derive an analytical expression for the peak amplitudes and show that the ratio of any two adjacent peaks is a function of driving amplitudes and detuning. This is attributed to the photon distribution in the coherent states and provides a measure for characterization of photon statistics in nonclassical coherent waves. We also demonstrate an Autler-Townes-like splitting of side peaks, the magnitude of which scales with the scattering order.