I recently got a video demo that shows a tool called The RingMaster. Their site: http://www.ringmastertool.com
Basically it's an attachment for your existing lathe, or you can buy it as a tool including the motor if you do not have a lathe already.

It allows you to cut rings from flat stock, which you then stack and glue to form bowls and vases.
The advantage of this is much less material cost/waste to make a bowl, plus it allows some creativity in assembling the rings - using different color woods produces interesting patterns on the sides of the bowls.

So I was thinking, how hard would it be to make something that would do the same job on the shopbot? Is the key to aligning and stacking these rings of wood, just the fact that they are cut on an angle?

If so, would it be possible to mount the router on an angle rather than 90 degrees to the table?
Then it's a matter of spinning the flat stock beneath the angled bit.

Alternatively, what about leaving the router as-is and making an angled turntable which the flat stock could mount onto?

This would produce the same result as tilting the router. To cut rings you would need to slowly lower the Z axis each time the angled turntable makes a complete revolution.

Or maybe this is too much fussing around...