In a triangle, any chain of circles that touch their neighbors and successive pairs of sides of the triangle counts at most six circles.
References
Evelyn, C. J. A., Money-Coutts, G. B., Tyrrell, J. A., A Theorem about a Triangle and Six Circles, in The Seven Circles Theorem and Other New Theorems, Stacey International, 1974, pp. 49-58.