Use the included link to determine which model G4 you have. Then see if it has a PRAM battery. If it does disconnect it and see if you can boot. If so you must choose between leaving it out and entering the date and time each time you boot or replacing the battery with a new one.
Before you mess with the PRAM battery, does anything happen when you press the power button, i.e. power-on noises, etc., or is it completely 100% dead? Also, what color is the light on the AC adapter plug? Try removing the battery from the machine, and see what color the plug is with the battery removed. If should be green. If it's amber, you have a bad DC-in, and they're integrated into the logic board on the Mercury model, so basically your logic board is toast...only the DVI version of the Titanium has a replaceable DC-in.