Largest hyperbolic disk embeddable in Euclidean 3-space? Hilbert proved that there's no complete regular ($C^k$ for sufficiently large $k$) isometric embedding of the hyperbolic plane into $\mathbb{R}^3$.  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 $\mathbb{R}^3$?
Edit: This doesn't seem to be getting many views, so I'll bump this by adding in a rather easy lower bound from the pseudosphere.  First, the pseudosphere is parametrized by the region $$\mathrm{PS}=\{z \mid \mathrm{Im} z \ge 1,\; -\pi < Re z \le π\}$$ on the upper half-plane model of $H^2$.  Let $z=x+iy$, so that ordered pairs $(x,y)∈ H^2$ when $y>0$.
Next, Euclidean circles drawn on in the upper half-plane model with center $(x,y\cosh r)$ and radius $y\sinh r$ correspond to hyperbolic circles with center $(x,y)$ and radius $r$.  I can fit a Euclidean circle of radius $π$ centered at $(0,1+π)$ into the region $\mathrm{PS}$.  This corresponds to a hyperbolic disk of radius $\operatorname{arctanh}(π/(1+π)) \sim 0.993$.
Surely one can do better?
Edit 2: fixed mistakes in formulas above (didn't affect the bound).  Here're some pictures:


 A: Several Russian geometers have addressed this question. I suggest you a survey on Isometric immersions by A. Borisenko (2001, I think) in Russian Mathematical Surveys (it is in English)
A: I can't comment on Noah's answer, so:
The reason for the $C^1$ condition is that Nash's $C^1$ embedding theorem says that any Riemannian $k$-manifold with a short embedding into $\mathbb{R}^n$ has an isometric $C^1$ embedding into $\mathbb{R}^n$ for any $n > k$. In particular, there is an isometric $C^1$ embedding of the hyperbolic plane into $\mathbb{R}^3$. The very beautiful proof proceeds by "pleating" the embedding to be closer and closer to isometric, while keeping it contained in a not-much-larger region.
A: I didn't see the exact answer to your question in the Borisenko paper, since section 2.4 only seems to address immersions of subsets of ℍ2 into ℝ3. However, a perturbation of the pseudosphere, Dini's surface, which is an isometrically embedded one-sided tubular neighborhood of a geodesic in the hyperbolic plane (see https://mathoverflow.net/a/149884/1345), seems to do the trick since it contains arbitrarily large disks in the hyperbolic plane. See Dini's Surface at the Geometry Center.

A: I think that you can get an arbitrarily large disk.  The proof is by crochet.  Since there's a pattern for crocheting constant negative curvature disks where you increase the radius as you go and since we live in 3-space, it follows that you can get arbitrarily large disks.
See this TED talk for some cool applications of hyperbolic crochet to biology, or this article for a more rigorous explanation.
