Well if you count movies then as far back as I can remember that would be spirited away or...ghost in the shell..one of them two. I was a bit hestitant of anime back then, I think I had only seen the very disturbing animes back then, so I was expecting spirited away to scare me xD But it's one of my most favourite movie's of all time now.
As for an actual anime tv show, probably Naruto, I was quite addicted, now adays I can't stand it.