But in reality, it's cryptographically checking the validity of bitcoin transactions. So the system churns for the benefit of the system itself, instead of churning for the benefit of some external goal.
And I don't think you should dismiss this criticism so easily. I personally have very little confidence in bitcoin as a currency. But if the system also contributed to science by folding proteins or finding Mersenne primes or doing anything else that is externally useful, I would consider participating.
Protein folding is a probably better example. Or, I don't know, modeling long-term weather patterns?
Here's a list to start with: http://en.wikipedia.org/wiki/List_of_distributed_computing_p...
The actual pointless task is to try to find a type of hash collision, to be specific: find a plaintext containing hashes of the transactions to be committed such that the hash of this plaintext starts with a bunch of zeroes. The more initial zeroes you require, the higher is the difficulty. (The plaintext is allowed to contain a small arbitrary nonce. So miners just cycle through all the possible nonces, computing hashes, until they find one that starts with a bunch of zeroes.)
It's conceivable that one could somehow mix the transaction data into a problem of bioinformatics, but I don't know so much about that. The problem is that, as far as I'm aware, some hash of the transactions to be committed must be part of the input of the "pointless" problem.
There are lots of problems in bioinformatics and physics that are hard to compute and easily divided into chunks. Find one that also has easily verifiable results and you've got a pretty good candidate.