A theorem usually attributed to Soddy (but I guess actually disovered 
originally by some unknown Japanese) says that given any two tangent 
spheres, internally tangent to a third, a chain of exactly six tangent 
spheres can be nested about the first two and inside the larger.

http://mathworld.wolfram.com/Hexlet.html

This is a picture from the simplest case, when the two original spheres 
are the same size, and the union of their collinear diameters forms a 
diameter of the larger sphere.  Then the six spheres in the chain are 
all identical and with diameters 1/3 of the largest sphere.  Again, 
rendered using the "reflective spheres and spotlights" trick.

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