When you draw a circle in a plane you can perfectly surround it with 6 other circles of the same radius. This works for any radius. What's the significance of 6? Why not some other numbers?

I'm looking for an answer deeper than "there are $6\times60^\circ=360^\circ$  in a circle, so you can picture it".

The short answer is "because they don't work," but that's kind of a copout.  This is actually quite a deep question.  What you're referring to is sphere packing in two dimensions, specifically the kissing number, and sphere packing is actually quite a sophisticated and active field of mathematical research (in arbitrary dimensions).  

Here's one answer, which isn't complete but which tells you why $6$ is a meaningful number in two dimensions.  The packing you refer to is a special type of packing called a lattice packing, which means it comes from an arrangement of regularly spaced points; in this case, the hexagonal lattice.  The number $6$ appears here because the hexagonal lattice has $6$-fold symmetry.  So a natural question might be whether one can find lattices in two dimensions with, say $7$-fold or $8$-fold symmetry, since these might correspond to circle packings with more circles around a given circle.  (Intuitively, we expect more symmetric lattices to give rise to denser packings and to packings where each circle has more neighbors.)

The answer is no: $6$-fold symmetry is the best you can do!  This is a consequence of the crystallographic restriction theorem.  The generalization of the theorem to $n$ dimensions says this: it is possible for a lattice to have $d$-fold symmetry only if $\phi(d) \le n$, where $\phi$ is Euler's totient function.

The generalization implies that you still cannot do better than $6$-fold symmetry in $3$ dimensions.  There are two natural lattice packings in $3$ dimensions, which both occur in molecules and crystals in nature and which both have $6$-fold symmetry, and it turns out that these are the densest sphere packings in $3$ dimensions.  It also turns out that they give the correct kissing number in $3$ dimensions, which is $12$ (see the wiki article).

In $4$ dimensions, the kissing number is $24$, and I believe the corresponding packing is a lattice packing coming from a lattice with $8$-fold symmetry, which is possible in $4$ dimensions.  In higher dimensions, only two other kissing numbers are known: $8$ dimensions, where the $E_8$ lattice gives kissing number $240$, and $24$ dimensions, where the Leech lattice gives kissing number $196560$!  These lattices are really mysterious objects and are related to a host of other mysterious objects in mathematics.

A great reference for this stuff, although it is a little dense, is Conway and Sloane's Sphere Packing, Lattices, and Groups (Wayback Machine).  Edit:  And for a very accessible and engaging introduction to symmetry in the plane and in general, I highly recommend Conway, Burgiel, and Goodman-Strauss's The Symmetries of Things.
