But what about a hybrid system where you could add an hour or two of usage in that first few seconds of charging?
Would be great during a layover in a typical US airport where you might have to sit on the floor in a corner to use a power outlet. You could plug in for 10s to fill the cap, eat lunch while the cap charges the battery for an hour, plug in again for 10s before you fly out having something close to a full battery.
A cellphone battery is say 1 Amp Hour. Assume an equivalent capacitor is 3600 Amps for a second. Assume the capacitor is charged to 10x the Voltage, and is charged in 10 seconds. That needs 36 Amps - copper wires a bit thicker than those in a kettle cord. Bad approximations made, but right order I think.
Also there are real safety issues with something that can be charged that fast, because it usually implies it can be discharged faster. A shorted Li ion battery may burn, but a low-resistance capacitor will cause an explosion. Capacitors are fun :-)
An iPhone battery is roughly 1400mAh at 3.7V, which gives 5.2Wh or 18720 watt-seconds. A standard power outlet in the US supplies 15A at 120V, for a maximum of 1800 watts. So if you could charge your phone using the maximum power available from a standard outlet, it would take roughly 10 seconds. A typical laptop might take about 1-2 minutes. (That's assuming 100% efficiency, which is unrealistic.)
Would be great during a layover in a typical US airport where you might have to sit on the floor in a corner to use a power outlet. You could plug in for 10s to fill the cap, eat lunch while the cap charges the battery for an hour, plug in again for 10s before you fly out having something close to a full battery.