>> you put a text file in Dropbox, and it showed up as a formatted page on the web
It's pretty easy to configure a generic Jinja2 HTML template, then run the markdown and Jinja2 python modules on any markdown file to turn it into a web page.
A framework is hardly needed, but these exist if you don't want to configure a solution:
Jekyll and Hyde are both awesome. I played around with Pelican a while ago, and it seemed solid.
Droptype wasn't much more than that. It used Dropbox's then-new revised API to look at Dropbox/Apps/Droptype and then displayed any Markdown files as a list. If you added a README file, it would render the contents of that. It was a quick and dirty Django app.
There wasn't any control over the design or any web-based editing capabilities. Others are doing interesting stuff in that space, most notably http://scriptogr.am/ which is probably a lot like where Marquee would have ended up if we continued to go keep going down that route. They're doing a very nice job with it though.
That said, we're looking at Dropbox/GitHub integration as more of a feature than a product. We're approaching the problem space from a somewhat different direction; the focus on our authoring environment is the first step to getting there.
A framework is hardly needed, but these exist if you don't want to configure a solution: