I wonder if it's possible to design (and build) a digital sundial with arbitrarily many digits (the obvious upper bounds of 'arbitrary' imposed by our physical universe notwithstanding). It's an interesting thought experiment...
I don't see why not, although limiting factors will be brightness and diffraction limiting how hard the edge of the shadow sweeping across the ends of your optical fibers can be and also how small the fibers themselves can be.
edit: spelling