Hilbert proved that there's no complete regular (C^k for sufficiently large k) isometric embedding of the hyperbolic plane into R^3. On the other hand, the pseudosphere is locally isometric to the hyperbolic plane up to its cusps (though it has the topology of a cylinder). What's the largest hyperbolic disk (with Gaussian curvature -1) that can be smoothly (or C^2, say) isometrically embedded in R^3?