You know, I'm not gonna try to avoid games others have mentioned, like I usually do. There's too many people mentioning too many games to do that. So without further ado...
Super Mario 3: Probably the best of the original Mario games on the NES. Probably one of the best Mario games ever. Even one of the best games ever. It's got solid controls. Well designed levels with tons of secrets. Some of the most varied and iconic power-ups of the franchise. It's the total package of a Mario game. A future Video Game museum would be committing a crime if this game was not included.
Sonic the Hedgehog 3/Sonic and Knuckles: Effectively the same game (especially if you know the history), I consider these the pinnacle of the Sonic franchise. Up to three playable characters who all feel unique. Great and useful power-ups like the fire and lightning shield. A variety of special stages. And some of the best levels and music of the franchise. The definitive Sonic game(s).
Final Fantasy VII: Not gonna lie. This is my all time favorite game. It's revolutionized the JRPG genre. And certainly helped put the genre on the map. I expect even WRPGs owe a thanks to this game. Plus, it has the insanely useful materia system, introduced the Limit system, and has iconic characters, villains, and scenes. Even players who have never played the game know the scene where Aerith dies or Sephiroth walks into the flames.
Final Fantasy Tactics: A must for Strategy RPG fans. I've yet to find a game with such a deep job system. Such solid game-play. Or just such a good story that breaks the mold. Rather than playing the hero that saves the world, you play as the man that history named a heretic. A man framed as a monster, who fought for a world that betrayed him, even if he only wanted to save his sister. The story alone is a reason to play it. Though, I fully understand people getting lost in it.
Uncharted 2: Part of the Uncharted franchise, in my mind this is the best. A perfect example of what cinematic games can truly achieve. This game feels like playing a classic Indiana Jones film. And it's got solid gameplay, characters, and comedy to boot.
Batman: Arkham Series: I recommend playing one of these games, the first or second if nothing else. These games show what a game about a Superhero can be like. It's a perfect adaption of the character, and makes you feel like the Dark Knight. Great atmosphere and well portrayed characters with iconic voice actors, this is a must for Batman fans. And a solid game to boot. Probably the best licensed games ever made.
The Last of Us: Yes, I know plenty of people have praised this game. Is it wrong to say that this game deserves the praise? I can say that the gameplay is pretty average, but the story and how it is presented is where the game shines. It's beautiful, I grew to love the characters, and the ending still leaves me wondering how I feel about it. Truly, few games have made me feel so much.