We investigate the possibility of electrostatic potential saturation, which may lead to the phenomenon of effective charge saturation. The system under study is a uniformly charged infinite plane immersed in an arbitrary electrolyte made up of several microspecies. To describe the electric double layer, we use a generic local density functional theory in which the local microionic density profiles are arbitrary functions of the local electrostatic potential. A general necessary and sufficient condition is obtained for saturation, whereby the electrostatic potential created by the plane becomes independent of its bare charge, provided the latter is large enough. As a consequence, for most situations, the following simple and practical sufficient condition follows: if, as the electric potential psi--> infinity, the local theory predicts that the highest valency counterions density diverges as psi(nu) with some nu>1 or faster, then the electrostatic potential will saturate. Otherwise, if the counterion density diverges as psi or slower, or does not diverge as psi--> infinity, the electric potential will not saturate. Using this condition, we investigate the possibility of the saturation phenomenon within the framework of recent theories proposed in the literature to describe electrical double layer beyond the Poisson-Boltzmann description.