When I was young, around five or six years old, my mother had a friend who was a writer who wrote about different cultures. One of those was Japanese. Once she let me see a cartoon she had. She did not call it anime but "Japanese cartoon" and that was the first anime I ever saw (Princess Mononoke). To me then it did not look like anything special. During my childhood I saw quite a few pop animes (which to me were no different from American and European cartoons), ones that almost everyone knows (Pokemon, Digimon, Shaman King, Yu Gi Oh etc.) and I even liked Digimon a lot.
I started to watch anime again around May 2012. The first anime I watched then was Angel Beats! and it really moved me as well as Clannad and some other I watched in the summer of that year (Death Note, Code Geass and a few more.) After watching these and the many more I watch to this day, I get the feeling that if I would have never known anime as well as I do now and if I'd never experience all of this, I'd probably be missing a part of my life.