To prolong a rechargable battery's life, is it better to recharge it constantly or first let it run down?

The best way is something inbetween. Overcharging a battery can damage it, reducing its life, so it is best not to leave it charging continuously. Basically, overcharging causes a build-up of crystals inside the battery. Completely draining a battery, however, reduces the batteries life as well. Use the battery for a while, then recharge it to full capacity before it is completely drained, but don't store it in a charger.

This is my cell phone..Don't they just shut off when they are done charging? Or do you have to take it off the charger?

Many cell phone battery chargers will shut off when the battery is fully charged. In that case, you might as well leave them in the charger. Not all do this though, so you may want to check the literature that came with it.

Also, if it is a lithium ion (Li-ion) or nickel metal hydride battery (NiMH), overcharging is less of an issue than if it is a nickel cadmium battery (NiCd).

ok..will do..thanks again Methos!