Is it possible to construct an incrementally accurate rectification of a circle? Exact rectification of a circle (construction of a segment exactly the length of the circumference of a given circle) has been proven impossible. There is a number of rectification constructions that give a pretty good approximation, e.g. Kochański's Construction, which gives 0,002% accuracy. Is there any construction though, that - following a repeating set of steps, gives arbitrary accuracy, depending on the number of repetitions? A geometric equivalent of a series convergent to Pi? (of course I mean the theoretical concept, it may not be practically useful - assume infinitely precise compass and ruler, and infinitely sharp pencil ;)

Sure :

Suppose you have constructed the regular polygon with $2^n$ sides. Then it's easy to construct the regular polygon with $2^{n+1}$ sides : just bissect each central angle. And we know how to construct a square.