Guybrush has a special ability, and at one point of the game you can use that ability, see what happens when you ‘use it up’.
Collect all the insults.
Don’t read any guides(!).
Open doors in Meleé Island(tm) (and see where you go)!
Look at the clock.
Find both(!) endings (save at the start of episode 3).
Discover Monkey Island(tm).
Find out what color a tree is if it falls and nobody is there to see it. (Actually, this is from MI2!)
Listen to the MT-32 and CD-Audio (and maybe the remake soundtrack).
Do what the game tells you to do at the end of the game.