[Hilbert proved](http://en.wikipedia.org/wiki/Hilbert%27s_theorem_%28differential_geometry%29) 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?