I think when discussing the term greatest, people often get it conflated with best. For instance, when it comes to playability, artstyle and so on, modern games are much better than older games. But when it comes to greatness, we have to look at things like how revolutionary it was, how influential the game is, and so on. On that end, I would say the greatest video game I've ever played is Legend of Zelda: Ocarina of Time. As for my personal favourites, I would say either Ōkami, Legend of Zelda: The Wind Waker, or Pokémon Crystal.
The Legend of Zelda: Ocarina of Time: It's not a very original choice, but it's one I think most people can agree has merit. It combined groundbreaking graphics, a high quality soundtrack, a wonderful story and memorable characters in one, and still has great replayability 16 years later. Perhaps it's just nostalgia, but the excitement of wandering outside of the Lost Woods for the first time and seeing Hyrule field was one of the greatest feeling I've had in my years of playing video games. From breaking into Hyrule Castle, the absolute skullfuckery that was the Water Temple, escaping Ganondorf's castle, Ocarina produced so many great moments in video game history it's hard to think of what could top it.
Ōkami: It's an artsy game for sure, but playing as Amaterasu, just exploring and running around Nippon was like jumping through paintings on my TV. The game was one of the few examples I can think of that truly justify video games as both a storytelling medium, as well as one for artistic expression. Not to mention you play as a wolf, what can get better than that?
The Legend of Zelda: The Wind Waker: My personal favourite in the LoZ series, the Wind Waker was a game that received a disproportionate amount of criticism when it first came out. People were complaining about how it was making the LoZ series too childish, too light and cutesy, but when it comes down to it, it's the one in the series that has held up the best. The cell shading style was unique for its time, and even today, the game is just really nice to look at, with the rounded styles and wavy lines. The games use of large bodies of water made it feel truly grand and exciting, like going on a true adventure.
Pokémon Crystal: I could just as easily throw in Emerald, Platinum or Soul Silver in here, but when it comes down to it, Crystal was the game that got me truly hooked on pokémon. I don't think it held up as well as my nostalgia tells me it did, but I'll be damned if I deny how much this game means to me. I remember cursing that infernal Miltank of Whitneys after it took out 4 of my mons, only for it to be felled by my trusty Bayleef and it's lucky critical Razor Leaf. I remember stomping through Morty with my Shadow Ball Eevee, laughing at Chuck as my Meganium effortlessly beat him, and the most iconic battle in the series: Red at Mt. Silver.