The capacitance of the double layer formed at a metal/ionic-conductor interface can be remarkably large, so that the apparent width of the double layer is as small as 0.3. Mean-field theories fail to explain such large capacitance. We propose an alternate theory of the ionic double layer which allows for the binding of discrete ions to their image charges in the metal. We show that at small voltages the capacitance of the double layer is limited only by the weak dipole-dipole repulsion between bound ions, and is therefore very large. At large voltages the depletion of bound ions from one of the capacitor electrodes triggers a collapse of the capacitance to the mean-field value.