This is going to date me a bit, but what the hell. In chronological order of games that blew me away and gave me an unforgettable experience:
1. Secret of Mana - I had played a few RPGs prior to this, but everything about this game blew me away at the time. The graphics, the style, the story, the fact battles were real time instead of turn based. This is one of the games that cemented gaming as a serious hobby for me.
2. Shenmue - going back and playing this game now it doesn't stand up, but at the time I had never played another game that felt like I had so much freedom. I could buy things from vending machines! I could fight, play, get a job, play arcade games, etc .... This was my first taste of the open world.
3. Skyrim - I missed almost all games between 2002 and 2010 because I was having real life adventures in China. Skyrim was a mind-blowing reintroduction to what gaming could be. I got completely lost in the game for almost a year after it released.
Close to making the list:
Ultima Online
Civilization II
Final Fantasy Tactics
Halo
Red Dead Redemption
Fallout 3
Witcher 3