Six episodes, plus maybe one extra, for your sampler disk to a newbie with the intent of getting him interested in the form.
I would tailor the sampler to the particular recipient. My friend Bill would probably be most interested in series with distinctive art, so for him I would choose such shows as Mononoke and Kaiba. John would be more intrigued by complex stories such as Serial Experiments Lain. Deborah has a taste for grand fantasies, so perhaps the Ah! My Goddess movie would appeal to her. And so on.
For a potential anime fan whom I don’t know, the following might might constitute a decent introduction to the charms and range of the medium.
Angelic Layer, episode one — Cute kids, dolls, fighting, high tech, problem families.
Azumanga Daioh, episode twelve — More cute kids, high school, sentimental comedy. (Better this Chiyo-centric episode for starters than the first, which has too much Tomo.)
Dennou Coil, episode one — Not-so-cute but very three-dimensional kids, affinities with both Miyazaki and Ghost in the Shell, high tech, mystery, humor. It also illustrates the shortcomings of the licensing non-system: it’s one of the best shows of recent years, yet it may never be legally available in region one.
Mushishi, episode one — For sheer strangeness.
Paranoia Agent, episode eight — For the exceedingly dark humor. (Kon’s series is for college-age or older viewers only. If the prospective fan is a youngster, substitute an episode from your favorite comedy.)
Seirei no Moribito, episode one — Fantasy adventure, court intrigue, a strong female lead, outstanding animation.
As an extra, I’d include a CD of music from Cowboy Bebop.
It’s impossible to represent all the salient aspects of anime with just six examples — there are no magical girl or space war shows listed above, for example — but these might give the viewer some vague idea what anime is capable of.
Oh yeah, there’s always the first episode of Haibane Renmei.
Addendum
Speaking of Dennou Coil: it’s Halloween weekend, which is a good time to mention Miss Michiko.