That big universe makes it an ideal place for fans to play (yes, I know that was an awesome segue). Fans who want to create their own stories within the Star Wars world have plenty of characters, locations, and timelines to choose from. I don't know if I could narrow it down. Lots of creative geniuses do though. Fan films abound. A quick and dirty search for fan films yielded over 8,000 results. You can find parodies, dramas, reboots, documentaries, and plenty of Stormtrooper humor. The Stormtrooper / Cops parody comes to mind.
The 2011 challenge is still open, if just barely. You have until 12 noon PST on 6/6/11. Read the official entry information on Atom.
Here's an entry for this year's challenge that I adore; it shows what commercials might be like in the Star Wars universe and features a cute, dancing power droid - like Gonk!
Unlimited Power by Eliot Sirota
And this video that shows the original trilogy in two minutes in paper just makes me smile (and marvel at the level of patience that other people have - how does someone cut that much construction without going bonkers?!). It's not entered in the fan film challenge.
Video by Eric Powers, song by Jeremy Messersmith
If you're in the mood for fan films, I have a suggestion for you: don't aimlessly click through YouTube results. You could find some gems, but other intrepid folks have done the hard work for you. Search for "best of" lists or the winners from the past eight years of fan film challenges. If you have a favorite, leave a link in the comments!