Skyrim was my Steam gateway drug. I had never been a heavy gamer before then, but oh my gosh, I was blown away by how beautiful and amazing the Skyrim world was. The mythology, the vastness, and the stories all of that and more just had me hooked.
The rest is history but two of my other favorite game series Dragon Age and Mass Effect have also had a big impact on me (both were also accidental discoveries, I found out about Bioware when Mass Effect 2 was free on Origin). I can't even begin to explain how much I love Mass Effect and Dragon Age. Ugh. The characters just worm themselves into my heart and then either rip it to tiny pieces or heal it or some combinations of both. The storylines and plots have really stuck with me from the moment I played them. That's quality writing right there. I feel like they've definitely aged well and will always have a place in my heart.
Those series really set a standard for me on choices matter, worlds, storylines, and characters. They're definitely the most influential games for me, even now.