Alternative economies within Bitcoin
Imagine you wanted to run your own fork of Bitcoin, for example as part of a next-generation Bittorrent protocol that discouraged freeriding.
Is there any way to do this while taking advantage of the high hashrate (and thus security) of the Bitcoin network?
It seems so. Imagine you start your own sub-economy with 1 Bitcoin. Because transaction histories are a matter of public record, anyone could "follow" that Bitcoin as it or fractions of it flow around to other accounts. Only descendants of that special Bitcoin would be treated as part of your sub-economy/currency.
You'd have to have some rules to disentangle the mixing of Bitcoins representing your currency with other Bitcoins. For example, you'd have to handle the case of an account receiving Bitcoins that are from outside your sub-economy, though it's not too hard to imagine what these rules would look like.
The transactions in your sub-economy would be in low Bitcoin amounts, so in some cases transaction fees would have to be paid to get them included in the block chain. If an alternative economy has low transaction volume, with high value, the users would be willing to pay these fees out of pocket. Or the users of your alternative economy could mine Bitcoins to offset the cost of the transactions. 
This is a bit of a hack, but I think it could work. 
-  In the case of using this technique to power a Bittorrent fairness economy, you'd want to use distributed coin flips with a low probability of a "payment" happening to pair down the transaction volume.
-  One could also imagine future crypto-currencies that explicitly support this functionality.