Any Fire Emblem game - Of the few turn based strategy RPGs I have played, to me, this series is the best of them all. Interesting characters, somewhat unique gameplay options (I have yet to see anything like support conversations, but if there are other games with a similar system, please tell me), and while the stories may not be unique, they are still pulled of well.
Any Zelda game - The Zelda series is the classic puzzle solving adventure game in my eyes. My most recommended game of the series is a Link to the Past, as I have yet to beat OoT.
Any Pokemon game - Seeing a trend here? All of these series have been around for a while, and there is a reason. They are all fantastic examples of their respective genres. I started playing Pokemon back when the originals were out, and I enjoyed it enough to keep playing even to today with the newest version I could get my hands on (Pokemon Diamond). I recommend Playing the newest or the Gold / Silver / Crystal generation. A fun little RPG, if a bit geared towards children.
Portal - Now this was an amazing game. Humorous, challenging, and very enjoyable. Short, but some of the most fun I have ever gotten out of my 360.
The Godfather - This may seem a bit odd, but bear with me here. This game is an example of a movie game done right. While it may not follow the movie 100%, I still greatly enjoy this game. This is the only game I have played where I can go from extorting a shopkeep to stealing a racket truck and using it to plow over a few civilians before bribing a cop and robbing the nearest bank. To me, this shows people that movie games can be good.
Honorable mentions:
No More Heroes - Fun to play, very interesting characters, and a funny, not quite predictable story all rolled into one awesome package.
Chrono Trigger / Cross - Two of the best RPGs I have ever played. Interesting casts of characters, excellent plot twists, and an engaging story.
Final Fantasy series - I have played most of them, and the ones I have played, I have loved (except XII, but it wasn't a bad game). Gameplay-wise, I likes FFX the most, and story-wise, FF VI. The first few are also classic RPGs.
Almost forgot about Mario. Mario games are the classic platformers, and everyone should play them at least once.
This list could go on forever, so I am going to cut it off here.