A lot of games have changed who I am, because they've been a large aspect of my life since I was a kid.
Final Fantasy VII is the obvious one. Aside from the fact that it's probably partially responsible for turning me into a dirty hippie environmentalist, it was also one of the first stories I'd encountered that taught me about death (I was seven when I was playing it, and hadn't experienced any death in my life). And, beyond that, it was the first time I ever wondered about concepts like identity - "Who am I? Are we just what we believe?" Yeah, even as a kid I was a real nerd. I thought about weird stuff. XD
StarCraft also had a big impact on me. I was playing it when I was about eight or nine, and, from that moment on, I fell in love with outer-space. I'm not kidding that, when adults asked me what I wanted to be when I grew up, my response was, "An inter-planetary anthropologist." That was my dream job in 3rd grade. =3 Unfortunately, the fact that we haven't discovered alien life kind of puts a damper on that. <_<
No, seriously, I don't think I could list all the games that have changed me in some way. I like to approach everything with the attitude that it will help me learn and grow. ^ ^