There is nothing special about light being reflected multiple times between two mirrors. Your sphere can be considered as 2 concave mirrors having the same radius and a common centre. In your case, the intensity of light will keep increasing until the mirror starts melting, unless it is perfectly reflective, in which case the intensity will increase forever.
In fact, that's how a laser works: a gain medium (that is, a medium in which stimulated emission can take place) is placed in between two mirrors. One of the mirrors has a very high reflectivity (as close to 100% as possible); the other one also reflects most of the light, but some of it goes through. A photon can bounce between the two mirrors thousands of times (and ideally stimulating the emission of other photons at each trip) before getting out through the second mirror.
Now, to put a spin on the OP, lets assume that the smaller sphere has the outside coated with a reflective layer and that the two spheres are concentric. In this configuration you’ll get a convex mirror in the centre and a concave mirror surrounding it. In fact, that’s one of the configuration used for lasers (have a look at the second figure on this page ) , so in principle you can get a spherical laser, provided that you fill the space between the spheres with a gain medium (e.g., a mixture of gases).
If you make the outer sphere semi-transparent, then your laser will look like a ball of light. If you make the inner sphere semi-transparent, then all the light will be focused in the common centre of the two spheres and you’d get a very intensity (which could be used for trivial purposes, like cold fusion ).
Also, from the wiki link above, it looks like someone managed to make a laser out of a drop of solution.
As for what's happening when light meets sub-wavelength structures... well, many things can happen :). As examples, you can have a look at [micro-structured optical fibres](http://en.wikipedia.org/wiki/Photonic-crystal_fiber) or even optical fibres having a [diameter inferior to the wavelength of light](http://en.wikipedia.org/wiki/Subwavelength-diameter_optical_fiber)propagating trough them. You get some interesting effects by illuminating semiconductor spheres several nanometers in diameter, aka [quantum dots.](http://en.wikipedia.org/wiki/Quantum_dot)