To murk the waters some more. The chemical reaction in a battery causes charges to build up on the anode and cathode. These "surface" charges act like a capacitor, and that's what allows current to flow instantly, like, well, a capacitor. You can measure this capacitance if you have an oscilloscope and an ac source. (Use an isolation transformer.) Connect a resistor between the battery and the source. Connect the scope in X-Y mode with the ground at the connection between the resistor and the battery. The angle of the major axis of the resulting ellipse is related to the value of the capacitance. The internal resistance of the battery may also affect the results. Anyone interested that has the equipment can experiment with batteries and capacitors to see what they get.