We report experimental and theoretical investigations of interacting excitons of the yellow series in cuprous oxide (Cu
2O) with principal quantum numbers up to by means of second harmonic generation (SHG). Using picosecond pulsed laser excitation up to 10 GW/cm
2 peak intensity we observe a pronounced change of the spectra with increasing pump laser intensity: an energetic shift to lower absolute energies and a spectral broadening. The absolute intensities of the spectral lines scale for low powers with the square of the pump power, but saturates at higher powers. At still higher powers the SHG intensity is actually reduced. To explain these results quantitively, we developed a semi-classical theory of resonant SHG where the process of SHG is fully coherent. The excitons are assumed to be bosons interacting by a distance dependent potential giving rise to both the changes in spectral line shape and the saturation by a
Rydberg blockade. The concomitant measurement of two-photon absorption allows to derive quantitative values for the exciton-exciton interaction. While the results agree in order of magnitude with those calculated by state-of-the art atomic-like van der Waals interaction theory, the scaling with principle quantum number is quite different. As a possible screening by an electron-hole plasma created by three-photon absorption into blue and violet band states could be ruled out, our results point toward fundamental differences between excitons and atoms.