My 2006 Yamaha Royal Star Tour Deluxe gives off a foul rotten egg smelling stench that seems to be coming from the battery. The bike has fried two brand new batteries in just two days of riding on each. What's causing this problem? Is it going to cost me an arm and a leg to fix it?

The egg smell is the battery. as it overcharges and overheats, this gas is flammable as well so no sparks.
The regulator is definately shot, but possibly the alternator windings as well.
Not certain on that model but it may have an alternator with an internal regulator.
Unfortunately can be an expensive exercise.