What is the largest possible thirteenth kissing sphere? It is well-known that it is impossible to arrange 13 spheres of unit radius all tangent to another unit sphere without their interiors intersecting.  This was apparently the subject of disagreement between Isaac Newton ("impossible") and David Gregory ("possible").  The cause of the dispute was, in part, that there seems to be a lot of room left over after 12 spheres.
How close of a call is it, so to speak?  What is the largest possible $r$ so that it is possible to arrange 12 spheres of unit radius and a 13th sphere of radius $r$ all tangent to another unit sphere, without intersections?  What does the optimal configuration look like?
 A: Pietro's version of this question is answered in a paper by Oleg Musin and Alexey Tarasov (to appear in Discrete & Computational Geometry, http://dx.doi.org/10.1007/s00454-011-9392-2, http://arxiv.org/abs/1002.1439). The configuration found by Schütte and van der Waerden (see Joseph O'Rourke's answer) is optimal and unique up to isometries.
The other version of the problem amounts to asking for the largest possible hole in a packing of 12 identical disks of radius $30^\circ$ on the 2-sphere.  I don't know the answer offhand.  One could certainly figure out what it must be by numerical optimization, but finding a rigorous proof would be difficult.  (It might be possible using variants of the Musin-Tarasov approach, which is an enormous brute force search over small planar graphs.)  I am sure someone must have looked at this problem, but I don't know of a place where the answer might be recorded.
A: The recent paper by Hopkins, Stillinger, Turquato,
"Densest local sphere-packing diversity. II. Application to three dimensions,"
Physical Review E 83, 011304 (2011)
(PDF link),
addresses the variant suggested by Pietro:

"The smallest radius spherical surface onto which
  the centers of 13 spheres of unit diameter can be placed
  is strongly conjectured to be $R = R_\min(13) = 1.045572\ldots$,
  with the centers arranged in a structure first documented in
  Ref. [5]. It appears that, although Gregory was incorrect in
  conjecturing $K_3$ to be 13, his guess was not particularly far off."
[5] K. Schütte and B. L. van der Waerden, Math Ann. 123, 96 (1951).

Here's a nice image (from MathWorld) that shows the gaps when 12 unit spheres,
tangent at icosahedron vertices, surround one unit sphere:
           (source)
Update. See Henry Cohn's answer, which cites a more recent paper that settles (positively) the
conjecture noted above.
