How about law? Obviously there are non-technical problems there as well, but it seems like there is still a lot that could be done technologically to improve transparency and (maybe more interestingly) reduce expense.
I'm a non-practicing lawyer in the tech space. There actually is a lot of technology in law. For example, eDiscovery is a really hot field right now. The basic problem: given a multi-terabyte corpus, which documents are legally relevant?
In the past one would gather a room full of lawyers and actually read through all of the documents (a task I did once). Now you can use a combination of linguists, lawyers, and search techniques to do the job.
(the discovery phase of a trial on average accounts for 50% of litigation expenses)
I've actually been thinking about law recently. It seems that, because it is a field that is based on rules, it would lend itself well to a more programmatic solution. Obviously not all law could be serviced programmatically, but there are certainly areas that could be.
There's already been quite a bit of work on applying technology to contract law relating to contracts (primarily, but not exclusively, financial contracts). There was a discussion about this on Lambda the Ultimate a while ago, if you're interested: http://lambda-the-ultimate.org/node/2973
