Many games have influenced me over the years, so its tough to narrow it down to a specific five. I'll try my best though.
1. Pajama Sam:
This game got me into gaming. I spent hours trying to solve the various adventures and puzzles on my dad's computer. After beating the game I replayed it many times because the game created a slightly different game each time. After playing it through many many times I began to discover every secret, every minigame, every dialogue. This game introduced me to gaming and got me addicted at an early age.
2. Super Mario Bros 3
After visiting a friends house I saw him playing Super Mario Bros 3. I had no idea how to play, but the game seemed amazing to me. Apparently the system was a bit outdated at the time, but I didn't care. Super Mario Bros 3 looked like the most fun thing I had ever seen. I didn't actually get to play, but it made me incredibly excited about gaming.
3. Super Smash Bros.
My grandfather purchased a N64, and my addiction began. I would visit his house at all hours of the day to play Super Smash Bros. with my friends. We mastered each character and had elaborate duels late into the day. Eventually we moved onto Melee which we still regard as the greatest multiplayer game today.
4. Puzzle Pirates
I became addicted to this MMORPG terribly, and still am. I have spent days upon days playing this game to an endless extent. However it is not exactly time wasted. From this game I have learned leadership skills, how to clearly teach people concepts, how to run a business and make a profit, and numerous other useful real-world concepts. My dreams have these puzzles in them, especially carpentry.
This game got me into the concept of games as a whole. I never finished a game, but I enjoyed every one. Playing with a group of friends or family for hours is an experience that modern electronic games have a hard time recreating.
Adventure/Third-Person Graphic Adventure
Role-playing/Persistent World Online RPG