DMC inspired me to look for the single most stylish and badass way to kill every enemy I come across in my games, I'll specifically look for a way which gives me a spectacular finish, like jumping off a ledge and sticking a blade in the back of a guy's head in Assassin's Creed, or launching a Warthog off a mancannon into my opponent on Halo.
It also made me a better gamer, the hair fraction reaction times you need to pull off some of the better moves in DMC have compelled me to train up, a skill which I've been able to take into every other game, Call of Duty for example, enemy jumps out on me and within a second I've knifed him in the face and moved on.
It's also given me an opportunity to look at games in a more in depth manner, taking the time to enjoy everything around me, from the scenery to the combat, as well as the story.
Honestly, I doubt I'd be the gamer I am today if I didn't pick up the original Devil May Cry all those years back.