Quite possibly this problem (or one like it) has been discussed before, if so I would be interested in any information about it. I just thought about it tonight, and so I haven't done any work on it yet, except to generate the image I will use to describe the process:
(I'm hosting it on my website to avoid the clutter that would be involved had I uploaded it elsewhere.)
Starting with four circles, each of radius 1, and centered respectively at (0,0), (2,0), (0,2), and (2,2), we have a framework for the addition of other circles of diminishing radii, which are all tangent to prior circles in the construction. Next is the circle centered on (1,1) with radius √2 - 1. Continuing, it is evident that the next set of circles (not shown) are four in number, and are each tangent to two of the original set of four and also tangent to the single circle of the prior generation.
Not only the circles and their construction are of interest, but the region of the domain lying outside all the circles would be, I assume, a fractal, and might have non zero measure and dimension.
Any pointers to "prior art" would be appreciated.