The Kepler-Bouwkamp constant is defined as the limit of radii of a sequence of concentric circles that are simultaneously inscribed in a regular n-gon and circumscribed around a regular (n + 1)-gon for n ≥ 3. The outermost circle, circumscribed around an equilateral triangle, has radius 1. We investigate what happens when the number of sides of regular polygons from the definition is given by a sequence different from the sequence of natural numbers.