In the crown ring I attempted to make, I used both gorilla glue & liquid nails. Didn't hold up well--the ring shifted with transport, storage, and drying out such that most of the glue didn't hold. I'd screwed all the pieces together (which is what held), but it was questionable.
Hide/animal requires good joints (makes great invisible seams). Titebond company makes a liquid version of it, but I've heard it's not as good as made from the dried granular form. You can get the granular form from a local luthier/violin repair shop for small amounts (they can also tell you how to mix it). I've heard if you know what you're doing, you can get the glue joint to crack open cleanly with hide glue.
Make sure your joints are tight. There are some photos of people making a laminated ring in the albums here. I suspect you could use a bunch of veneers or really thin plywood.
You could use one glue (hide?) for the laminations then cover the thing in an epoxy to seal it, too.
