Suppose you start with two points in the plane which are distance 1 apart, which for concreteness can be $(0,0)$ and $(0,1)$. Then you keep marking new points based on ruler and compass constructions. That is, you start with those two points as "marked. " At each stage you draw all possible circles which have a previously marked point as a center and with another previously marked point as a point on the circle, and you draw every line which goes through any two marked points. Then you mark every point which is an intersection of a circle or line and a circle or line. The set of marked points is the set of "constructible" points and has been studied since the ancient Greeks.
Here is a variant of that. Instead what happens if at each stage instead of throwing in every new line and circle, we only throw in those from a single pair of points chosen randomly and uniformly from all currently marked points? That is, suppose that at each stage we pick two marked points chosen uniformly at random, and we throw in the line and two circles determined by those points, and then mark every new point resulting from those circles and lines intersecting with each other or our existing circles and lines.
Question: With probability 1 do we still end up with all constructible points in our resulting set?
This seems hard. I don't even see any obvious way to prove that with probability 1 our resulting set is dense in the plane (which is trivially true for the set of constructible points). Based on the answer to [this question], one can choose specific points to get very far away from the origin very fast but there one is carefully choosing specific points.