That's an interesting question! We need to consider more than the area of the plate (we need two plates). Your general idea is that a battery could be exhausted by charging a very large capacitor and that is true.

If a capacitor is charged by a battery and the battery is disconnected, a charge remains on the capacitor. In working on electrical equipment, you have to be careful about putting your hands around large capacitors even after the power has been shut off. You can get a shock from them. When you get such a shock, you are getting hit by a transient current. It's the discharge current instead of the charging current.

By the way, I found a site that has the equation for the transient current when a capacitor is charged.or discharged It is

Capacitor Discharging.
The definition of "current" is that it is the rate of change of charge with respect to time. Batteries are usually rated in "amp hours". To answer your question quantitatively, we would have to figure out the total amp hours used to charge a large capacitor. Doing this for the transient would involve calculus. Have you studied calculus?.