Just speaking out loud here, not from experience. For reference where I work we use SVN for all of our projects, with the exception of kernel/u-boot work which is kept in git and patches stored in SVN.
I am a programmer, my wife is a graphic designer. I love revision control. She hates it.
What about having a system that autonomously monitors the live drop box folder. As files are changed / added / removed it keeps a secondary git repo ( separate from your code repo ) up to date. With the right tools on the server you can download a snapshot of all assets a any point in time without having to clone a terabyte of history, but the history is still ( mostly ) there. You won't get perfect history ( for example if a file is renamed ) and you run the possibility that a file is updated twice before your local copy is synced so you miss the intermediate. Maybe drop box has API's that would give you this information.