Dark and Hero Chaos
Only one kind of Chao is immortal: the Chaos Chao. If you have a Chaos Chao he will never die and reincarnate. Creating a Chaos Chao is probably the hardest part of raising Chao, aside from making an all S rank Chao.
There are three types of Chaos Chao.
The Neutral Light Chaos dates back to SA1: it looks like Chaos and, instead of a regular ball, it has a shining light above its head (which is where the "Light Chaos" name comes from).
The Hero Chaos is an angel Chao with pink and white features and an iridescent halo.
The Dark Chaos is a devilish gray-black Chao with large horns, a ball of blue fire above his head and a fire effect on his extremities.
All Chaos Chao are wingless and have a semi-transparent effect on parts of their bodies. They do not have facial expressions.
Dark and Hero Chaos
Firstly, your Chao must have died and come back twice. That's the long and hard part. It doesn't matter what your Chao was in either of those lives, but he must have reincarnated two times before he can become a Chaos Chao.
Once you have a baby Chao who fulfils that requirement, give him one of every kind of animal: twenty-one in total. Do not give him more than one of any animal, and do not give him any Chaos Drives.
Then you wait. If you want a Hero Chaos, pet and feed the Chao with a Hero Character; if you want a Dark Chaos, use a Dark character, and if you want the original Light Chaos, keep him neutral.
When the baby Chao evolves, you'll hear creepy organ music instead of the usual jingle.
If you did muff it somehow, your Chao can still try again next life.
Chaos Chao look gorgeous and make nice trophy Chao to show off, but they cannot mate and so represent a kind of evolutionary dead end. Don't do this to a Chao you think you might want to breed from later.