That's a pretty hard question, medias influence their watchers in an often unpredictable and undetected way.
But if I had to pick one anime, I would say Parasyte. Throughout the show, the parasites raised some interesting points which wouldn't hit the same if it came from a human, someone the viewer would apply "common sense" to.
Skipping the specifics of the show, it basically made me see things without perceiving them as right/wrong or at least not as much.
How it influenced my life: I've gotten less anxiety of making mistakes and being perceived as a jerk. For example, I used to very often make any excuses to leave when I'm near someone that knows me. But now, I might just shamelessly be there even if I says nothing meaningful. It's not that I'm going to all hangouts or stay until the end though.
This might be quite a long stretch but I don't know about philosophy or that thinking stuff over the show, lol.