Run out of electrons
A battery is a device which has a bunch of electrons on one side of it and a lack of electrons on the other side. When you put a wire between the contacts, the electrons run to the other side until there is charge balance. Then, you're right, no more electrons will flow. You have to recharge the battery or get a new one. Note that even when you "run out" of electrons in the charge balanced case, all of the electrons are still there, they're just in a different position. Some are left in the wire; most are back in the battery just in equal amounts on each side of the battery.

In short, you're right though.

With AC power, you're pushing and pulling electrons 60 times a second in order to extract work out of them. In that case, all of the electrons are always there and don't really move unidirectionally unlike the battery case.