Right now I know of 4 options:
- CloudHQ is expensive, but does it very well.
- Some desktop apps do the trick (such as Insync, by far the best option I know of).
- Zapier, IFTTT and cloudwork only go so far - they sync new files, don't keep synced after they change.
- Manually downloading files as PDF (or DOCX, XLSX, etc) might still be the simplest free reliable option.
Unfortunately I could find no Google Docs Connected App that do the trick, and dropbox have no app market, just their API, to which nobody found a way to do it properly, other than cloudHQ, as far as I know.
A good option to using freaking MS DOC format is going with http://stackedit.io or something similar such as the awesome penflip. But there's nothing as easy to collaborate with than Google Docs, with other non-tech-savy people.
I don't think there's any free way to automatically sync docs and dropbox without a machine.