In this Brief Report, we calculate the electric-double-layer (EDL) electrostatic potential in a system of several layers of immiscible electrolytes. Verwey-Niessen theory predicts that at the interface between two immiscible electrolytes back-to-back EDLs are formed. The present analysis extends this idea to the case where the immiscible liquids are contained inside a domain with given electrostatic potentials at its boundaries, where the thickness of the individual liquid layer can be comparable to the EDL thickness. Such a system gives rise to a situation where the overall EDL electrostatic potential in the system is dictated by the competitive influences of the boundary-induced effects and the effects induced by the jump in the ion-solvent interaction potential at the liquid-liquid interfaces. Invoking Debye-Hückel linearization, we derive an analytical result for the EDL electrostatic potential for two immiscible electrolyte layers, and extend it for a general system of N such immiscible electrolyte layers. We demonstrate that, depending upon the nature of the interfacial ion-solvent interaction potential jump, the overall EDL potential may manifest a strong influence of the boundary conditions or may invert the influence of the boundary conditions. Effects such as a variation of the ratio of the permittivity or the thickness of the liquids also dictate the overall potential profiles.