I can never pick just one of anything, and I'm not sure that I would call any of these my absolute favourites, but they do deserve to be more widely known:
-Fire Tripper is a one-shot time travel adventure/romance OAV that proves that Rumiko Takahashi can write a serious story with an ending if she wants to. VHS release by US Manga Corps back in the day, but I'm not sure it ever made it to DVD.
-The Cockpit is a three-part OAV about WWII from the Axis side (but overall disapproving of what happened on that side of the war, and the segments tend to end in tragedy). Based on a Leiji Matsumoto manga, and as often happens with his work, there's a lot more there than is initially obvious. Not light entertainment. Apparently there was an Urban Vision VHS release, but no English-language DVD (so says Wikipedia).
-The Sensualist is an odd historical OAV based on a Japanese novel of the 17th century, The Life of an Amorous Man. Despite the title and the presence of on-screen sex scenes, I don't remember it containing anything that would make it more than R-rated. If you're looking for action or much in the way of plot, though, this isn't the one for you—it's more of an anime art house film. No North American release ever, and very difficult to find generally.
-And since I've name-dropped a few serious ones, let me round out the list with a silly one: Special Duty Combat Unit Shinesman . . . but only the dubbed version. If you don't crack up at some point around, "I'm gonna die in a Godzilla suit!", you're probably not human. ;)