I have a piece of code written in Python. I would like to put that code in a webpage. Brython seems like the simplest way to glue the two things together, but I don't have a server that can actually run code on the server side.
Does Brython require server-side code, or can I host a page using it on the cheap with (say) Dropbox?
Brython itself seems to be completely client side, but whether that will be enough really depends on the code you wrote. It is not a full blown Python interpreter and does not have the libraries. You might want a backend to support it or use another client side solution as suggested in the comments.
Given how few real web hosters support Python, I think it is very unlikely that Dropbox would be suitable for this, in case you do need processing on the server as well.