Mine involves a time machine
Although many people like to have a conversation about what the "best" game of all time is, the conversation about the most "important" game you've ever played in your life doesn't happen with the same frequency. Some games just come along at the right time in your life, whether it's adolescence or your journey into young adulthood, and influence you in ways that no other game has before it.
Let me explain mine to give you a bit more background on what I'm talking about, so you can share yours in turn.
The most important game I've ever played in my gaming career is Chrono Trigger. Although I had dabbled in many JRPGs before it such as Dragon Warrior and a select few games in the Final Fantasy series, Chrono Trigger arrived at a perfect time in my life. I was approaching my teenage years, and had more time to play games so long as my schoolwork was finished -- this allowed me to become engrossed in the genre like never before. Naturally, I rushed home to finish all my homework so I could play it as often as I could -- if there was a time to do the "act like I'm going to sleep then jump out of bed and play the SNES with the volume off" trick as a kid, this was it.
I was completely swept away by its world, characters, and narrative, and couldn't put it down. It was one of the first games I felt compelled to beat multiple times over right after finishing it, which is something I had never done before with an RPG. It was also one of the first games that really "enlightened" me in terms of learning how to play JRPGs when it comes to tactics, and the general rules of the franchise (grinding, dedicated exploration for best-in-slot super-items, and so on).
Chrono Trigger is far from my favorite game -- it's not even my favorite JRPG (Skies of Arcadia holds that honor) -- but it holds a special place in my heart for a lot of other reasons, and I'll never forget it.
Has any game influenced you in similar ways?