This configuration seems to be Coxeter's loxodromic sequence of tangent circles. According to Wikipedia:
The radii of the circles in the sequence form a geometric progression with ratio
$$k=\varphi + \sqrt{\varphi} \approx 2.89005\ ,$$ where φ is the golden ratio. k and its reciprocal satisfy the equation $$(1+x+x^2+x^3)^2=2(1+x^2+x^4+x^6)\ .$$ The centres of the circles in the sequence lie on a logarithmic spiral. Viewed from the centre of the spiral, the angle between the centres of successive circles is $$\cos^{-1} \left( \frac {-1} {\varphi} \right)\ .$$
Higher dimensional generalization was done by Coxeter (1968).
