Anything that needs immutability and decentralization. You're asking for non-financial examples, which are not immediately apparent to me. Say, for example, you wish to make an online encyclopedia but you don't want to have any main server, instead you want lots of nodes that mirror one another, and that way if one fails the others work. However, you want to make sure that the edits are cohesive and people don't just delete articles or put their own on, so you add a protocol (like a cryptographic hashing function and consensus algorithm) so that additions to the articles happen together across all nodes, and every node has an easy way to make sure that the "fingerprint" of the latest edits matches the ones from all the other nodes. I must admit, blockchains are good for this sort of project, but the need is particular and creative. It would be like making a fully transparent application where the database is visible by all. Not every application wants or needs this, but some things, like scientific journals, online encylopedic archives, ledgers, research that is timestamped, copies of financial reports, many things can benefit from living in blocks that are cryptographically hashed to ensure their consistency across mirrors and also ensure their cohesiveness between changes.

Ultimately, you can have something like github be a completely decentralized blockchain based service, but it's not exactly necessary, and although it may be more elegant, it's not easier to debug, but it does make additive collections of human knowledge easier to condense, branch off, extend, and reason about.

It's not about one being infeasible to make and the other being feasible, it's about eliminating points of failure. If I have one long cylindrical vertebrae and I break my spine, I'm Effed. If I have many vertebrae, each can take its own shock without really compromising the rest. Can you think of anything that needs decentralized points of failure and impeccable record keeping? Again, the only things that really come to mind are things humans are trying to tabulate or keep adding to (knowledge and research, transaction ledgers, what else?). Most of these characteristics are fully attainable with standard databases that do atomic writes and have append-only architectures. It's an overloaded term because people think blockchains solve problems databases don't. It seems to be a very specific use-case but in the future it is likely we will use such tech to note our "observations" and ensure that they are cryptographically inconvenient to alter. For example, we could take all of twitter, hash it, store it, use it as a starting point, and years from now (assuming Quantum Computing doesn't destroy all encryption right out the gate) we can positively assert that some specific thing was said at some specific time on twitter because the past events in the twitter blockchain we just made would be increasingly difficult to alter. I guess that took me a long time to bring to words: anything you need the historical data to be unalterable for is a good use for blockchain.