Try the very first interaction of the movie. Elijah Wood and, forgive me, Ian McKellen don't do relaxed laughter in any way convincingly.
Yeah, after a bit I found several (yes I actually put them in and watched them sped up), and in the extended cut a few more.
Gandalf and Frodo when "A Wizard is never late", Gandalf again laughs when the fireworks go off for the children chasing his cart, an extra Hobbit pouring a cup of ale, Frodo pushing Sam to dance with Rosie, in Rivendell the Hobbits laughing and hugging after Frodo is well (see Return of the King for similar), Gimili over Legolas' suggestion about getting him a box to see, and can it be called laughter Merry & Pippin greeting Aragorn, Gimili, Legolas ect at Isengard? and then there are a few times you might add an orc character, Gollum, and the Meat's back on the Table, Boy's line outside of Fanghorn.
So yeah I think its safe to say I missed some.
In the extended Cut there is a brief laughter in the Tavern in Hobbiton, when Merry & Pippin flip Boromir on his bum, and Gimili during the drinking Game.