That's exactly what Sahil Lavingia is planning to do starting on Sunday, February 7th. Lavingia is the developer behind Color Stream and the Twizzle Twitter client [iTunes Links], and he'll be posting updates to keep us all apprised of his progress at his One Week App blog.
On the blog, Lavingia will be posting screenshots of his coding, timelapse movies (sorry, no streaming video of Lavingia at work), and other notes about his work as he moves forward. Lavingia is a realist -- he knows that he could create an amazing new app, or he could fail to finish his work in the allotted time. We don't know what he's planning on coding, either. Perhaps it will be a game, maybe a design tool, or something else altogether.
If you are fascinated with what happens behind the scenes in the app development process, be sure to follow Lavingia's one week app experiment.