Somewhere down the line, someone will formally or informally do a background check on you. You don't want to give someone a reason to subtly backstab you later. If you do something big on the way out, it can haunt you later, and you may never find out. Be as professional as possible, document everything you know that may help someone, and work full days during your notice period.
Now that I've said all this, you asked for stories...
- A former colleague gave the minimal possible notice "because every day I stay costs me a lot of money." He was leaving to start a t-shirt company. (He was a great guy, and not as arrogant as the story sounds.)
- A friend at an investment bank (which was notorious for perp-walking people out when they were let go during the financial meltdown) quit, and gave his notice from his new firm's Bloomberg account.
These actually sound fairly lame. I wonder if ohters have better. :-)