I need to address the fact that there are so many different video games that involve monkeys (or any similar primate). The first step is to try to name all of the games, or characters that I can possibly think of (obvious or not):
* Monkey Island - mainly because of the name
* Super Monkey Ball
* Marvel vs. Capcom 2
* Donkey Kong (DKC, DKL, DK64, blah blah blah...) - including Diddy, Dixie, Cranky, etc...
* Spanky's Quest
* Animal Crossing
* Gordo 106
* Ape Escape
* The Legend of Zelda
* Samba De Amigo
* Contra: Hard Corps (prehistoric secret ending)
* Kung Fu Chaos
* Metal Gear Solid 3 & 4
* Aladdin - Abu
* Chulip
* The Lion King - Rafiki & the gorilla/ape-thing (hakuna matata stage), plus many more monkeys in level 2 and later on in the game
* Mega Man series - Mecha Monkey (Monking/Monkikki), Spark Mandrill, Aperoid/Big Monkey, Hanumachine, Data, & Buster Rod G
* Spyro the Dragon
* Bio Force Ape
* Congo's Caper
* Super Mario 64* Congo's Caper
* The Adventures of Darwin
* Yoshi's Island
* Rampage
* Zack & Wiki
* Time Splitters
* Dragon Ball
* Primal Rage
* Star Fox - Andross & crew
* Monkey Magic
* Son Son
* Breath of Fire II
* Monkey Hero
* Pokémon - Mankey, Chimchar, Pansage, etc...
* Hail to the Chimp
Okay, I'm going to stop here. You are welcome to add to this list in the comments section, if you wish.
Now, I want to discuss two of the best monkeys (IMO):
#2 - Spanky (from Spanky's Quest for the SNES/SFC & GB)
Natsume does not disappoint with this action/puzzle monkey game. The soundtrack is so freaking tight, especially in the SNES/SFC version! It was so good that I couldn't leave it alone, and was basically forced to cover one of its songs. So, Spanky himself is a cute little monkey that has to attack fruit, and things by bouncing different sized and colored bubbles off of his head, which then explode into a specific sports related ball depending on which type of bubble you set off. He also gets to wear some silly hats that give him various helpful abilities.