The Death Egg is a large Death Star that floats around. It was made by Dr. Robotnik, Robotnik Sr. and Dr. Eggman while they were plotting against Sonic. It now resides at the bottom of the ocean (Thanks Sonic!).
Once morning, Dr. Robotnik got out of bed, and decided he and his father were too big of losers to beat Sonic on their own. They sent ten thousand robots to track down Dr. Eggman. Dr. Eggman thought this was an attack, so he sent his own soldiers. They fought for three weeks before they realized it was all a mistake.
The three of them got together, and started discussing a super weapon they could make. Of course, Dr. Robotnik fell asleep for most of it, and Robotnik Sr. was off at some random party, so Dr. Eggman was on his own. Dr. Robotnik snored a lot, and this made things difficult, but a mallet seemed to do the trick. Now, Dr. Robotnik can't snore anymore. Amazing what one good face-pounding can do!
When Godzilla was injured during the Ultimate Showdown of Ultimate Destiny, Dr. Robotnik allowed Godzilla to live in the Death Egg, using its life support system. When they tried to use the Death Egg, but Sonic blew it up. Dr. Robotnik fell asleep again, so Dr. Eggman jumped into the ocean, but then Dr. Robotnik woke up, thinking the thing in the water was Sonic, and shot every missile he had at it. Dr. Eggman survived, but Dr. Robotnik and Dr. Eggman never worked together again. Godzilla was merged with the Death Egg and destroyed in the blast.
Dr. Robotnik rebuilt the Death Egg, and claims that it's all his, not Dr. Eggman's.