I can't really think of any video games that profoundly impacted me in ways that books, movie, and TV have, but there are a lot of games I've really enjoyed and have stayed with me over the years.
Games
Grim Fandango
Over a decade since playing this I can still say that this is quite easily one of my all-time favorite games and one that I like to replay every year or so to remind me why video games are still worth playing. A colorful cast of characters, all of whom have a personality and reason for being in the game, with a great take on the film noir genre and Aztec mythology.
Silent Hill 2
This one is pretty non-explanatory if you've played the game, but if not here's a basic rundown. It's not a happy game, not in the slightest. It's not even a very fun game, with clunky controls and outdated graphics, but man does it ever hit you in the gut and toy with your perception of the world.
Half-Life 2: Episode One
While episode two definitely has the more emotionally powerful ending, I really enjoyed traversing City 17 with Alyx and her talk of zombines and the like, and found myself trying to get back to her as quickly as possible when we were split up, not because I felt I needed to protect her, but because I enjoyed being around her and felt safe with her, like nothing could touch me. Essentially I was experiencing the emotions commonly felt in the first few weeks of a romance, the constant need to be around one and other.
The Legend of Zelda: Majora's Mask
I've always held this game in high regard, but it wasn't until recently that I really discovered why. The complete feeling of hopelessness at the game's situation juxtaposed by the little feelings of happiness when I complete something like a temple, then the feelings of hopelessness as all my accomplishments are erased when I have to restart the cycle. Coupled with the darker music and some of the interesting theories about the stages of grief and you get an incredibly deep game and one of the best things Nintendo have ever produced.
Film
The Shawshank Redemption
I cry at the end of this movie every. single. time. I know it has a happy ending but it doesn't matter, it still makes me cry, which is why I really only watch it once a year or so, just to remind me of what a great movie it is.
My Neighbor Totoro
I ended up watching this with my niece when she was 4 and I was 22 and I have to say this is a movie that really penetrated any cynical/pessimistic shielding I wrap around my body and just made me feel happy. It's one of my nieces favorite movies and I watch it with her almost every time I visit my sister. If/when I have children, this movie will most certainly be a part of their childhood.
Toy Story 3
This one pretty much goes without saying, if you were a child of the late '80's or early '90's this is a movie that really got to you. My friends and I had to make an agreement before we watched it that if anyone of us made fun of another for crying, that person would be buying the drinks for the rest of the night. Those last 15 minutes man... I really do try not to cry in public places, but I couldn't help myself and just let loose, fortunately pretty much everyone else in the theater was doing the same thing so it really wasn't so bad.
Books
The Long Walk by Richard Bachman (aka Stephen King)
I read this one when I was 13 and it really got to me, like in a way that Stephen King's other works never had. I think it was me just trying to wrap my head around this idea of why young boys would voluntarily join a contest that would most likely end with their death, or how soldier's could willingly shoot them. It was a novel that, when I was done reading, I needed to go on a very long walk myself and just... try to process what I had read. Very few novels have had that effect on me, but this is just one of them that stayed with me long after it was done.
Trainspotting by Irvine Welsh
I cannot think of a better book for a misanthropic teenager than Irvine Welsh's 1993 novel Trainspotting, a book that one review said it deserved to sell more copies than the Bible, a quote I still agree with. What the book does better than any other book I've read before, is teaches you how to stay misanthropic while still finding ways to justify it. After reading that book you could be experiencing the greatest thing in the world and could still find its faults or how it's nowhere near as great as many think. It's a book that steals a little piece of your soul, and can really change your life (for the worst) if you're in a vulnerable state like say... adolescence. Proceed with caution...
Exalted by White Wolf Publishing
I cannot think of another RPG series that has had a more lasting impact on me than White Wolf's magnum opus Exalted. As much as I love their World of Darkness material, Exalted is still my go-to RPG for creating high profile campaigns and introducing new players to the wonderful world of pen and paper RPG's. I still enjoy D&D on occasion, but I think Exalted is superior to it in every way possible minus the pretty broken combat system that leaves you rolling 300 D10's for a single Charm. Despite the game's mechanical flaws, the setting of Creation has enveloped me in ways no other fantasy series ever has.