People interested in this idea might also be interested in Hashcash. Someone who wants to send and email to someone else also has to provide a "proof of work" with their message, essentially the same thing as Bitcoin mining. The point of it is to reduce email spam by attaching a small cost to each email, since, today, there is no cost difference between sending 1 email and 1 million, which enables spammers to send out so many.
The main benefit is that no real currency is ever involved, which reduces a lot of administrative overhead.