internet file system
Mobile apps have already made steps from "we will save your data on your phone" through "save your data in our cloud" to "save in your cloud, i.e. Dropbox". Apparently even the filesystem API change in the latest Android OS has the same goal: make using the remote storage transparent. Now there's one more step left: stop supporting a list of cloud storages, start supporting all of them. Though you'd need a universal protocol for that and I have found one.
remoteStorage is an open protocol for per-user storage. It aims at both the private clouds like ownCloud on your home NAS and at big farms. You can think of it as a combination of several known technologies, but it also adds an important part of being universal. And these have already proven themselves: REST+CORS for file manipulation, OAuth for limited privileges, and WebFinger to configure the whole system just by providing an email address.
And obviously it doesn't stop at mobile apps. A promising possibility is serverless web-apps (aka unhosted). Well, you just need to somehow provide HTML+CSS+JS and then they work with users' own storages. And even quick growth won't cause scalability issues.