I have read noumerous threads and watched many videos on combining Dropbox and Git functionality. One such thread is this: Using Git and Dropbox together effectively?
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I really don't think Dropbox is a good solution for this. For what you want, I would probably git clone
to a USB stick and carry that around.
Barring that, all the online git hosts that I know of (github, etc) allow you to download a tarball of whatever commit, and many of them allow you to edit files in-place.