I think most websites can be generated statically. But yes, it does mean a lot of the existing patterns used on the current web needs to be redesigned for it. Again, this is all very new so there's a lot that needs to be worked out.
I can give an example of non-static data (though what is static vs dynamic can be a grey area):
SPAs work great with Beaker/Dat since users can download the app and use it offline. The data can be any Dat archive. So for a social network, each user can have their own Dat archives of images and posts. The root site can hold an index of each user and download individual files from their Dat and display them using client-side routing. In this scenario, each user has their own database as a Dat which is indexed by a parent Dat website.
Dats aren't static. They're public key addressed, so you can make changes. The next protocol iteration will have support for multiple writers using CRDTs (sometime this summer).
Myself and others are currently volunteering to help bring the Dat Archive API to Bunsen Browser, a mobile Dat Web client currently only for Android (unless someone wants to jump in and make the build for iOS).