Hi there, you ask interesting questions. I have two things for you that aren't exactly what you're looking for, but they are definitely relevant.

First of all, a while ago I made a serverless chat for in your browser. It's just a proof of concept, it does work however. There will be nobody there, but you can test it by opening two browser tabs or using a second device. It works in the same way that Skype works, peer to peer. It's cool, because it is in the browser. http://codepen.io/Azeirah/pen/BHnbz

Second of all, you know bitcoins, right? It has this blockchain thing. People are starting to realize the potential of the blockchain. I've seen multiple initiatives and examples of applications using the blockhain to store the data without a static database. Note however, the data is still stored somewhere, on the users' pc's!

This is not a database-less application, it is decentralized. It's still very cool however, and may have a huge influence on the internet in the future if it stabilizes.

https://eris.projectdouglas.org/ (and their github: https://github.com/project-douglas/eris)