For instance, I want to be able to host my own files without having to make some sort of deal with my ISP. Although I've never had a problem serving requests directly from my computer, it's usually against the rules of ISPs. With a regular internet connection, I should be able to more easily store files on my own hardware and be able to share them with people as easily as I could with Instagram. Maybe there could even be an always-on hardware device that integrates in an open way with third parties that can serve those files, but ultimately those files live on that hardware and can be designated their own domains for URL resolution. Peer-to-peer and Cloudflare support could come with it out-of-the-box to mitigate load issues.
Meh, maybe nobody wants anything like that. I still like the idea.