If my transponder key battery dies, can I still start my 2005 caravan?

The battery in the fob is used to power the transmitter for remotely opening doors, trunk, and horn/lights (for when you lose your car in a parking lot). The transponder chip in the key does not require a battery. You can still start a car with a dead battery in the fob, so long as you still have the master key. In your case the fob and the transponder key are mated, but in fact the transponder chip does not require the battery and will still function just fine if the battery has died. The fob should take a standard watch battery.