AIM used to ask for a cryptographic checksum of a randomly chosen byte range of the AIM executable. The Gaim (now Pidgin) developers had to set up a server that would return checksums on demand. This doesn't meet your requirement of the verifier needing a small key.
Given that Dropbox apparently has no qualms about perjuring themselves in order to stop Dropship (or, as discussed in an earlier thread, lying about their security measures in order to defraud their customers), they could probably also take retaliatory action beyond just denying service to the user. Here are some possible retaliatory actions they could take:
* they could publish the user's private files, or simply look through them for the user's credit card numbers.
* they could sell the above data to the highest bidder.
* they could use it to attempt to impersonate the user to their bank in order to empty their bank account.
* they could randomly corrupt the user's data. (This might require a backdoor in the client software.)
* they could wait for an unusual volume of requests from the user (perhaps indicating that the user was trying to restore from backup) before terminating the user's account without warning.
* they could carefully comb through the user's files looking for evidence of any crime (illegal drug use, underage drinking, copyright infringement, possessing seditious literature, importing obscene material, defaming Islam, apostasy, embezzlement, tax evasion, whatever is the biggest no-no in their locale) and anonymously tip off the appropriate authorities.
* they could insert faked evidence of such crimes into the user's files, and then tip off the appropriate authorities.
Perhaps potential Dropbox users ought to be wary. This is a second data point in the company's history of seriously unethical behavior; one hopes they wouldn't engage in any tactics like the above in a dispute with a former user, if their extremely polite requests failed, but my experience is that people who behave unethically in medium-large ways often behave unethically in larger ways as well.
Caveat utilitor.