There's a pretty good example of a week-long enterprise in the book "Freakanomics." It's in a chapter called: "How is a prostitute like a department store Santa?" Levitt talks about how both of them depend on seasonal work. Santas only work during the Christmas rush, and the Chicago prostitutes he followed did most of their work during the week that Chicago hosts an annual convention for local politicians. Says something about the mores of local politicians, huh?
There is a yearly golf tournament not far from our house every year. Many "businesses" spring up for the week long tournament each year. Inevitably, many of them revolve around parking. Some people will sell spaces in their driveway or yard. There's a local group that lines up cars in a near-by soccer field and then offers a shuttle ride for a minimal fee. Some people will drive large vans up and down the side streets offering shuttle rides (for a fee of course). Depending on the weather, other small opportunities arise. If it's hot, I've seen people selling water and lemonade. If it's raining there are usually a few people selling umbrellas.
In the area of a sporting event, you could also host and run a sporting event like a 5k run, selling entry to the race and offering a prize to the top three finishers. On the day of the event you can do some vending of shirts and refreshments.
Another idea is to hold a gallery art show and sell art for just a week. This would need to be heavily promoted to work, but could make some money. The virtue of this idea is that if you make the right kind of arrangements with the artists involved there will be no overhead. The art that doesn't sell can go back to the artists. You don't have to buy it all up-front, but work on a commission basis instead.
I think mwalter has it. A business which focuses upon a short function (like sporting events) would fit within the constraints needed. Right now, one could look at a business for the Olympics (given their start last night). In order to be successful, you would need to find out other short-term businesses which have "sprung up" given the event.
I would suggest selling t-shirts of the Queen jumping out the of helicopter or replicas of the bronze leaves used to hold the Olympic fire.
I would suggest something that revolves around an event, such as the Super Bowl or the presidential election or something along those lines. It would make sense for the enterprise to be short-lived in that case.
I am not sure if this would count, but what about a booth at a fair? You can create a project for selling there, and just have the business while the fair is open. Another optioin is a fundraiser, because that is something that could last one week. Maybe you could rent a food truck or a mall booth?
These are all great suggestions, but can you maybe think of small-scale businesses, suitable for a 'got-no-cash', ordinary teen, still within a week?