To really answer this question I think two things need to be addressed. One is, what purpose are you showing anime to your friend for? To get him hooked? To just say, “Oh, this is anime?” To introduce him or her to the history of anime?
As many of people have pointed out, anime is really a medium, rather than a genre per se. It’s a bit like saying, what is the definitive “movie.” There are many subgenres, including sci-fi robot movies to romantic comedies.
So one question I think is worth asking is: When we think of an anime that can “represent” the medium, what qualities are we thinking about? Is it huge eyes and robots?
I would argue that visually distinctiveness is important; and you would need to show something that shows that range. Background score is also key.
I would also argue the complexity of character and storyline, and the ability to go beyond what we think of “normal” in a Western storyline is also key. So whether it is the blurring of good and evil (that’s why I would pick Nausicaa of the Valley of the Wind); the fantastical element; the looser boundaries of sexuality (Ranma ½), that’s quite anime.
I also think the there tends to be subversion of genre. Actually I think that’s quite Japanese – they take some invention from another culture, and they make it their own. You see this, no matter in writing, food, products (e.g., cars). They took the Western face and made it their own. They took the cowboy genre and made it their own. Even Evangelion, for example, is a subversion of the robot genre.
Finally, there’s cutting edge anime and there’s run-of-the-mill anime. Sailormoon, for example, might be “typical” teenage anime fare, but I really wouldn’t show it as representative of what anime can do.