I came in contact with 'Japanese cartoons' at the age of 10 or 12. I do not remember it very clearly, because my first 'love' was Dragon Ball and Dragon Ball Z. (At the same time a certain Hungarian TV channel aired Sailor Moon, Candy, and some other long forgotten anime.) Dragon Ball (+Z) is really long, so I do not remember clearly when I got hooked on them. Only years later did I realized, that the cartoons mentioned above were called 'anime'. I do not know whether it is worth to mention that the TV channel, that aired anime did it in a block from 16:00-18:00 like this: They first aired Dragon Ball, than Sailor Moon than two French live action series for teens. Also: I likedDragon Ball very much, but I hated Sailor Moon as much as I liked DB. This all happened during approx. 1997-1998.Dragon Ball Z was removed from the TV because 'It has too much violence in it.' (I was really sad. Also this was the time when everybody who could draw nicely began to draw Son Gokus day in and out.)
I began to watch animes again during High School (in Hungary it means form the age of 14 to 18). Maybe I was 16 when Spirited Away and other movies from Studio Ghibli made me in love with anime again. This time I was aware that they were called 'anime'. Also this was the beginning of my downfall. A classmate of mine liked anime as well and he introduced me to other titles. At the same time a new TV channel was born: A+ (today it is called Animax). The early 2000s were the era of Ghost in the Shell and Full Metal Panic! and a lot of other very good animes (e.g.: Inuyasha, Yu Yu Hakusho, Yu Gi Oh GX, Ergo Proxy, Paradise Kiss etc. the list is infinite).
I saw that somebody records every 50th anime they watched. I think that's a nice habit, so I tried to trace back my every 50th anime:
Some mix ups are sure to be there, but this was the best I could conjure up, as my memory is not soo good to remember all the little details.