An interesting fractal construction comes from rolling circles inside a stack of larger circles. This has been considered here and here... however I don't think these designs are the correct maths for proper conformal rolling circles. The equation I uses gives a different result.
For a set of nested circles of power 2 diameters, rolling with a constant roll contact point speed, in alternating directions I get this:
It is part of a set of fractals which are defined by the curvature ratio from parent to child. Here we see the family for ratios -4,-3,-2, 2, 3, 4. (-1,0 and 1 are not fractals).
