It's probably one of my favorite things.
We never actually stay until 3am, but we do stay until ~2am.
The Friday after Christmas usually sees Magic Kingdom staying open until 3am. We plan on sleeping late this day and maybe heading to EPCOT in the afternoon. At night, we'll head over to MK right before the fireworks (~11am.)
As a PP mentioned, the crowds are scarce around 1:30am. It's really pretty cool - especially when it's very crowded at WDW. Imagine a packed Main Street at 11pm for the fireworks. It's cool seeing it empty at 1:30am. Not to mention - there's some great photo opportunities. Here's some pics I managed to snap during late-night EMHs..