We report results of current versus voltage measurements on GaAs doping superlattices with current flow parallel to the superlattice symmetry axis, i.e., perpendicular to the layers. The material was grown by low-pressure metalorganic chemical vapor deposition and characterized by excitation intensity-dependent photoluminescence spectroscopy. The luminescence results show the superlattices to be of high quality. Four samples with nominally identical doping densities and layer thicknesses but different total numbers of superlattice periods were investigated. The electrical measurements demonstrate the strong dependence of the current on the number of superlattice periods. Our results agree semi-quantitatively with a simple model which attributes the current to thermionic emission over the multiple built-in space-charge potential barriers.