Pick up your favorite framework in the open-source community and contribute to that - either via code, tests, documentation or helping users. This you can continue for weeks and will help you in the long run.
Start a startup. Pick a little idea, build it, market it and see what happens. You'd probably learn a ton more that way than working remotely for 3 days.
You won't be able to learn too much in 3 days, and also none of the guys at a startup will take a lot of interest making you understand things considering the short tenure.
Pick up a small idea - some pain point that you see in your daily workflow. There are so many possibilities with free infrastructure today, Heroku, Google App Engine, and of course Amazon WebServices.
Start contacting people to see if they are interested and go through the proper way how someone seeds a startup. This will help you learn something that no experience ever will.
From a founder's perspective they lost the time. Probably for mitigation they would have someone else implement the idea too... which would indicate that they might not provide the same attention to your implementation.