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I've recently learned about Amsler's surface, a surface of constant negative Gaussian curvature. If I understand things correctly, there is a whole family of such surfaces, differing in the angle of intersection for the two lines that generate it. But I guess I'm only interested in the most symmetric version, where these lines are orthogonal. I'd like to know how large a portion of the hyperbolic plane I can embed on this surface.

So if the unit of length is chosen such that the Gaussian curvature of the surface becomes $-1$, what is the radius of a circle which is centered at the intersection of the asymptotic lines and just touches the cuspidal edges of the surface? A “circle” here would be the set of points on the surface with fixed geodesic distance to a central point. That intrinsic circle would not be a planar circle in the 3D embedding of the surface.

The following illustration of Amsler's surface was taken from the Gallery of pseudospherical surfaces by A. Ovchinninkov. I'm not sure how accurately it matches what I ask for, since there are other figures on the web which look somewhat different from this.

Fig. 14 of the referenced document

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The previous answer points to the paper, by Gemmer and Venkataramani [GV], which tells us that the radius requested is $1.2654$. But the previous answer makes some claims about the GV paper that might mislead.

The GV paper shows that hyperbolic disks of arbitrarily large radius can be isometrically and real-analytically immersed in 3-space. But the immersion described in [GV] is not injective for large disks, so it isn't an embedding. They immerse a large disk by letting it wrap many times around the pseudosphere.

This lack of injectivity can be easily fixed, however. Dini's surface is a twisted analog of the pseudosphere that is simply connected, rather than being cylindrical. So a hyperbolic disk of arbitrarily large radius can be isometrically and real-analytically embedded in 3-space as a disk-shaped subset of Dini's surface. Note that, with this technique, large disks will be wrapped, many times, very tightly around the screw axis of Dini's surface.

The previous answer also reports on results from [GV] about periodic Amsler surfaces. It is important to realize that the periodic Amsler surfaces defined in [GV] are only $C^1$ smooth; they fail to be $C^2$ along the joint where two pieces meet. This lack of smoothness is not a big issue for Gemmar and Venkataramani, who are actually trying to model how real-life thin disks will crumple when forced to assume negative curvature. It's not clear how interesting a $C^1$ embedding of a disk would be to a mathematician, however, since the entire hyperbolic plane can be isometrically embedded in 3-space, if we are willing to settle for just $C^1$ smoothness.

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this question was studied in . The figure above was taken from Amsler's original paper and the dashed curve closely matches the largest geodesic circle cut from the Amsler surface.

Interestingly, the authors show that arbitrarily large isometric embeddings of the hyperbolic plane exist. They show this by taking an Amsler surface with an angle of $\phi_n=\frac{\pi}{n}$ between the two generating lines and periodically extending the surface bounded between these two lines. However, it is still true that there does not exist an isometric embedding of the entire hyperbolic plane.

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