  A drawing of a graph in the plane is a thrackle if every pair of edges
intersects exactly once, either at a common vertex or at a proper crossing.
Conway's conjecture states that a thrackle has at most as many edges as
vertices. In this paper, we investigate the edge-vertex ratio of maximal
thrackles, that is, thrackles in which no edge between already existing
vertices can be inserted such that the resulting drawing remains a thrackle.
For maximal geometric and topological thrackles, we show that the edge-vertex
ratio can be arbitrarily small. When forbidding isolated vertices, the
edge-vertex ratio of maximal geometric thrackles can be arbitrarily close to
the natural lower bound of 1/2. For maximal topological thrackles without
isolated vertices, we present an infinite family with an edge-vertex ratio of
5/6.
