a simple scoreboard is easy to do if you wish to just use ajax and you can write a cgi-script.
Get your C2 game to make a request to a cgi-script which includes the score and name (or identifier) in your http call code
for example, the following link hits a perl file which is my cgi-script.
http://111.222.333.444/cgi-bin/add_scor ... &name=jack
after that you can have your cgi-script to read a text file (or xml file). process the new score, then write out the new file. your game can then use ajax to pull the updated file.
this will do what your post-title requests, however it does not do what your text says... you will need something that can run a cgi-script and that tends to need a server of some type. I don't know but I doubt dropbox will execute stuff for you.