Good heavens, what to recommend? How to keep it short? Let's try:
Without duplicating anything (that I know of--someone else already said Tampopo!) or getting too close (Princess Mononoke v. Howl's Moving Castle, for instance - both Miyazaki films) or getting too popular (Highlander). I'll try to give 4 or 5 of each.
Movies:
-Night of the Creeps - Do you like Horror? Teen slasher flicks? Cop drama? Mad scientists? Monsters? Suspense? Comedy? Space aliens? Zombies? Well, with one movie, you can have it all. Enjoy.
-36 Chambers of Shaolin (Freakin' HUGE influence in my life as I live it, though I've only ever seen it once, and as a pre-teen!)
-Holy Mountain - Absurdist cinema, very 60's-70's, but also very full of imagery and metaphor
-Bruce Almighty - Jim Carrey wackily delivers a surprisingly sobering cautionary tale about what can happen when you try TOO hard to get what you want. The message is far more than the movie itself.
-Who Framed Roger Rabbit? - The movie that changed at least one person's life. It's a landmark. You should see it. It's where the phrase 'bumping the lamp' came from. And it has Christopher Lloyd. And pretty much every WB and Disney character of any consequence.
Books:
-"The Belgariad" - 5 books by David Eddings, some pretty terrific high fantasy, a truly epic series.
-"Midworld" - Alan Dean Foster (one of many books set in the same universe) - I'm sure at least one of the folks at Avatar must have read this somewhere along the line. Also, "Nor Crystal Tears" is possibly the best first-contact novel I have ever read.
-"Tik Tok" - John Sladek - Imagine a robot experiencing a Johnny-5 sort of incident, except that he turns out like Bender. Only much more so. You might also consider "Adam Link, Robot" - by Eando Binder, who is a character very much like ST:TNG's Data, only Link was created way back in 1939.
-"The Man Who Never Missed" - One man decides to make a plan to Make A Difference. And he does. Oh yes, he does. Surprisingly relevant for being 25 years old.
-Any of the Callahan's books by Spyder Robinson. Time travelers, witches, musicians, love, spacemen, nuclear weapons, and some of the best Irish coffee to be found anywhere.
Music:
-Search for "Dr. Demento" and listen to whatever comes up. This may take a while. I'll wait. No, keep looking.
-"Hand Built by Robots" - Newton Faulkner
-"Further in Time" - Afro Celts (and Robert Plant, and Peter Gabriel)
-"Music for Peasants" - Urban Farmers - I can almost guarantee you won't be able to find this one, but if you do, it'll be worth it.
-"Wild Serenade" - DuOuD - Middle-Eastern instrumental music played electronica-style. It's actually pretty cool.
Television:
-Twilight Zone/Night Gallery/Tales from the Darkside/Monsters (I tend to lump them all together)
-Tiny Toon Adventures/Animaniacs/Pinky and the Brain (likewise)
-Special Unit 2 - didn't run very long, but you gotta check it out. If you like it, look into Psi Factor as well.
-St. Elsewhere - Before there was House, Chicago Hope, or ER, there was St. Elsewhere. Plenty in there to soak up.
Games:
-Shadow of the Colossus/Ico - These games know where your chains are, and pull them plenty.
-Morrowind
-Myst/Riven (Feel free to play Exile and the rest if you want, but make sure to catch the first two)
-Any of Microsoft's "Age of ..." games. They're not Starcraft, but they're still plenty entertaining RTS/Resource games.
--Patrick