What is the standard location for the "recent files" information that each app seems to store? The app in question is called RStudio.  When I am using it, if I want to look at a file that I have looked at recently, I can click:  File...Recent Files...and a menu of the last 5 or 6 files used shows up.

I want to sync this activity across devices using Dropbox, but I need to know which directory/file to add to my syncList to do so.  Any ideas?
The short answer is it depends... most apps will store it in ~/Library/Preferences/.  For example, on Mavericks the Preview recently used list is in ~/Library/Preferences/com.apple.Preview.LSSharedFileList.plist

You can use the terminal/commandline plutil to see what's in there: plutil -p ~/Library/Preferences/com.apple.Preview.LSSharedFileList.plist

If you want to get into the nitty gritty of modifying the file (outside the scope of this question, but what the heck), man plutil.

For syncing this file, you may find that there is other config info in there, I'm really not sure.  You'll want to make sure it's sane to do before pulling the sync trigger.  I'd suggest manually copying this a few times to make sure you understand what happens to the app if its files change while it's still running, etc.

I think you'll definitely want to stay away from syncing the whole directory since it would be possible to get into a loop where each application is trying to write to the file and dropbox is non-stop trying to propagate the changes across as each app competes for what's "correct."