Banjo-Kazooie/Tooie.
I remember my Dad got me Banjo-Kazooie for my birthday when I was a kid. Bloody loved that game...Though I didn't complete it until later on because I couldn't get enough Jiggy's to get past the 'Halloween level' (and I was afraid to do the actual level as a kid xD)
I don't remember when/where I got Banjo-Tooie, I just know that I own it

I think it was my Dad again.
The characters and levels are initially what got me hooked since everything was so humor filled and almost everything had a personality. It's one of the few games I can think of where even the collectibles and inanimate objects had a personality! That as well as the Mumbo's Magic with almost every level. I looked forward to seeing what I would be turned into next!
To me, Mario 64 laid the groundwork for 3D platformers, but RARE came along with Banjo-Kazooie and perfected it. With the setting, level design, power ups, abilities, Mumbo Jumbo animal transformations, humor, characters, bosses and of course, solid controls and platforming.
It's not just my favorite game on N64, it's one of my favorite games ever. Easily in my top 5.
I cry every time I remember that Micro$oft now owns RARE, especially when I look at how they're used these days (or more accurately, how they're
not used)...
- - - Post Merge - - -
There are the amazing Xbox 360 ports which are the defacto versions of the game, as they added in so much stuff from the original.
The ports are exactly the same as the N64 versions except the HUD is now in HD and certain parts had to be changed for copyright, such as the N64 logo walking in at the start...Which is now a few seconds of awkward silence because
laziness.
Oh...And the addition of those cringeworthy (and unnecessary) little cutscenes at the end advertising Nuts & Bolts.
They didn't really add anything that wasn't in the original game.