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][1] for some cool applications of hyperbolic crochet to biology, or [this article][2] for a more rigorous explanation.


  [1]: http://www.ted.com/index.php/talks/margaret_wertheim_crochets_the_coral_reef.html
  [2]: http://www.math.cornell.edu/~dwh/papers/crochet/crochet.html