In this work, a unimodular random planar triangulation is constructed that has no invariant circle packing. This disputes a problem asked in [9]. A natural weaker problem is the existence of point-stationary circle pack-ings for a graph, which are circle packings that satisfy a certain mass transport principle. It is shown that the answer to this weaker problem is also false. Two examples are provided with two different approaches: Using indistinguishability and finite approximations