[euge]

And it depends why and who your audience is. A crowd of beer drinkers might not realize that it isn't perfection (if it falls short of our expectations), but then again it might be just right for the situation.

If I'm not happy with it, I have a hard time giving it to someone else.

Those are words to live by for sure!

[JKL]

It doesn't have to be extract. But it's quick and easy.

I brewed my 1st all-grain 8 years ago and served it in 11 days for a Christmas Party.  4 days in primary, crash cooled and cold  conditioned in a keg for a week.  Sad to say it was one of the best beers I've brewed to date.  Even more sad is the notes I took while brewing it.  I've tried to recreate the beer twice with no luck.  All I know is 2-row, crystal, and victory malts.  A weird combination of Cascade, Fuggle, and EKG's every 15 minutes, and WLP023 Burton Ale yeast.  Maybe someday I'll try again?

[Malticulous]

Pitch a lot of yeast cold and ramp up the temp as fermentation progresses. Give it a day or two attenuation/diacetyl rest after fermentation appears to have stopped. Crash cool with finnings for two to three days then bottle or keg. I do this with a 1.047 Cream ale and it's ready in less then two weeks with bottle conditioning.

[abraxas]

My light amarillo wheat pale (~4-4.5%) was very drinkable if not best at two weeks.  I pitch a good amount of healthy, high flocculating yeast (Scottish), keep it cool for a day or two and then bring it up to about 70F for a week.  Crash cool for a couple of days, transfer and then carb in the cooler. 

If I had to I bet I could use the following schedule successfully:
Day 1: Brew and pitch yeast (keep cool)
Day 3: Raise to 70F
Day 5: Crash cool
Day 7: Transfer and force carb using 20 minute shake method, let settle for a few hours and drink

In place of a highly flocculating yeast, filtration or fining agents would be another option, I see getting the yeast out of suspension as being very important to a quickly turned-around beer.  I'd also keep the SRM levels low as in my experience darker beers need some time (phenolic reactions and settling?).

I think a 10 day beer contest would be fun.

[markaberrant]

The keys are
- pitch lots of healthy yeast
- inject lots of O2
- start fermentation cool, then ramp up

Once I started doing all of these things, my beers started tasting better AND much sooner. 

Aside from that, choose a lower gravity recipe.  Select a yeast that either flocs well, or choose a style where a bit of haze isn't an issue.

I find all beer will taste yeasty when fermentation first finishes... duh, there is a ton of yeast still in suspension.  Then for a few days (up to a week, sometimes two), it will taste very grainy/doughy.  You can somewhat mask this with a flavourful yeast (ie; weizen), or lots of late/dry hops.