Bill Gosper wrote:

Mandelbrot calls this Cantor set "Apollonian gasket", and wrote several years ago that its fractal dimension remained unknown.

People who like Apollonian gaskets should look at recent work of Jeff Lagarias and his collaborators (on the arXiv). The article that appeared in the Monthly ("Beyond the Descartes circle theorem" by Lagarias, Mallows, and Wilks; math.MG/0101066) would be a good place to start. A particularly nice new wrinkle of Lagarias' work is the notion of supergaskets: these don't exactly match the description

Then the distinct images of C under G cover C (except for a Cantor set) with non-overlapping circles.

(the complement isn't a Cantor set) but a supergasket does in a weaker sense "cover" the interior of the disk; more precisely, a supergasket extends a gasket to a dense family of circles, no two of which intersect in more than one point. See math.MG/0010302 .

Plotting the circles to high order [Thanks to the kind auspices of Stephen M. Jones, the circular gasket is http://gosper.org/suds.gif, the strip is http://gosper.org/stripcheese.gif

What are those colored circles? (I see yellow, purple, green, blue, and red when I look at this picture closely.) Are they added just to give the cheese extra pungency?

Problem: is there a regular expression (or even a subgroup) which generates each circle only once?

It's worth mentioning that the circles have unique representatives in a tree that generalizes the Stern-Brocot process. The construction for moving down the tree is a swap operation: given four mutually tangent circles, replace one of them (call it C) by the unique other circle C' that is also tangent to the other three. If you let the four initial "circles" be the x-axis, the line y = 1, and the circles of radius 1/2 centered at (0,1/2) and (1,1/2), then you get the gasket in the unit strip. If you refrain from swapping out the x-axis, then you just get the Ford circles, and the swap operation is essentially just the mediant operation (where the mediant of two reduced fractions a/b, c/d is (a+c)/(b+d)). Just as the Stern-Brocot process gives you a tree in which every rational number in [0,1] except for 0 and 1 occurs as a node in a binary tree, this "Descartes-Soddy swap process" gives you a tree in which every circle in the gasket in the region bounded by the four initial circles occurs once as a node.

maps the gasket onto the strip between the real axis and imag(z)=1, with the circles now including the Ford circles:

|z - n/d - i/d^2| = 1/d^2, for every reduced rational n/d.

Looks to me like there's a factor of two missing here. (The big circles that fit between the real axis and imag(z)=1 have radius 1/2, not 1.) I've fallen foul of this particular factor of two in the past. I'm not sure whether it's best to use the strip of thickness 1 or the strip of thickness 2. In any case, it's noteworthy that for all such gaskets, if the radius of each of the four original circles is the reciprocal of an integer, and you do swap operations, every derived circle will also have its radius the reciprocal of an integer. (This follows from the Descartes relation on the curvatures of four mutually tangent circles, along with the fact that if a monic quadratic equation in one variable has integer coefficients, and it has one integer root, then the other root is also an integer.) Jim Propp