I have had some that were great after 5 days. My most recent batch had a nice hiss and a great head at 5 days but was actually fairly flat when drinking. It's been 2 weeks and that batch has been spot on for several days.
3 weeks will be ideal for carbonating and conditioning, but if you don't have a pipeline running you will be anxious to drink those beers... That's fine, they will be carbonated long before that, but they will only get better as you go and the last bottle will be the best one. I'm working on getting my pipeline full by brewing 2-3 times a month so I can let each batch age a bit longer before getting into it.
Bottles: Weedwhacker Wheat, SimCity IPA, MO/Cascade SMaSH
Primary: Sasquatch Imperial Stout, Bee Cave Kolsch
Next to brew: Some sort of enamel-melting IPA and a nut brown