The two sequences in which you try to steal the idol in the first place. Much of the action takes place automatically in a room that you can't see; all you can see is dialogue and the actions that Guybrush is doing, many of which you can't normally do in the game. The highlight is perhaps "Hypnotize quarrelsome rhinoceros".
Repeatedly escaping from the vegetarian cannibals' hut causes them to increase security until the hut has a steel vault door, yet the obvious escape route is apparently not noticed.
The revelation that Herman actually has a ship.
The realization that "The governor can take care of herself" is more than just an excuse not to go rescue her...
The underwater idol puzzle, possibly the most hilarious example of Cutting the Knot in all entertainment. Guybrush has been dumped off a pier, tied to a heavy golden idol. You have ten minutes to solve the puzzle before Guybrush drowns, one of the rare genuine deaths in a LucasArts adventure game. There are swords, scissors, knives, razors, shears and all manner of sharp cutting things just out of arm's reach. If you wait long enough, two guys walk onto the pier, lengthily debate tossing a sword into the ocean directly over your head...then decide against it and walk off again. The solution, ladies and gentlemen: pick up the idol and walk out with it.