Wow, this is HARD. I like about every genre except sports games, so it's difficult to reduce it to just 5.
1. Teenage Mutant Ninja Turtles (Arcade) - side scrolling beat 'em up
2. Castlevania: Symphony of the Night - action/exploration
3. Bioshock Infinite - story-driven FPS
4. Final Fantasy VI - RPG
5. Vampire Savior - fighting
A few of my all-time favorites, like Q*Bert and Shadow of the Colossus, don't fit in a list like this because they're so unique they don't really represent a specific genre.