Here is a game for any plural number of players.
Needed: Blank piece of paper, compass.
First, with the compass, draw two circles of the same radius, with each circle passing through the other's center.
Players take turns. On a player's turn, he/she draws a circle (with the compass, of course) of any positive radius such that the center of the circle is at the intersection of at least two previously-drawn circles. The number of intersections passed through by this new circle is added to the player's score -- where each intersection is of at least two previously drawn circles and of the circle just drawn by the player, for a total of AT LEAST THREE circles per intersection (including the circle just drawn by the player). Note: The number of intersections (each of any number of circles), not the total number of circles in these intersections, is added to the player's score.
No two circles may have both the same radius and same center. (No two circles can coincide completely.)
The first player to reach a predetermined number of points wins.
(I suggest a higher goal score for a larger piece of paper and larger initial circles. I suggest a lower goal score if there a more than just a couple players.)
Note: If 3 or more circles ALMOST intersect at a point -- but their exact point of coinciding is uncertain because the circles were poorly drawn -- use geometric theorems to determine if indeed the three or more circles coincide *officially* at that point.