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.)
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 :-)