I want a national voting system with distributed ledgers in every state, and to add to that, every state has a copy of every other states voting records by virtue of having to validate the transactions. There's obviously lots of particulars around this, such as what to do with absentee ballots that are mailed in. This is a hypothetical use, at least in my mind.
Another one, more for the finance industry. A few years back, I paid my rent with a moneygram. A few weeks later, I got a phone call from the rental office telling me that moneygram rejected the payment and that I needed to call moneygram. Upon calling them, they said that they had record of me buying the moneygram (or whatever the hell it actually is), but that their system didn't have a record and there wasn't much they could do beyond give me a refund. How in the actual hell does a payments company have a problem like that? As a developer, I'm left to wonder if this was caused by some errant production release that wiped records, or a developer who ran a script from a well-known runbook that wiped out critical data (read: Amazon S3), or maybe even fraud somewhere in the pipeline. Why don't they have this data distributed and backed up in offline systems? If the government is willing to throw the book at financial fraud crimes (Unless you are Jamie Dimon), then there aught to be some kind of mandate that we have a verified ledger of electronic financial activities. The even scarier thing, to me, is that I thought they would have been doing that already...
Those are a few potential uses from my perspective.