Geodesic currents form the completion of the space of closed curves on a hyperbolic surface. The notion of geometric intersection number carries over continuously to this completion. Among its many interesting features, plenty of geometric structures on the surface can be represented as geodesic currents, such as hyperbolic metrics (Teichmueller space) or more general negatively curved metrics. These are examples of filling currents: geodesic currents that have positive intersection with all other currents. The subspace of filling geodesic currents can be endowed with a natural distance, generalizing a classical notion of distance in Teichmueller space. In this talk we study its coarse geometry. This is joint work in progress with Jenya Sapir.