We demonstrate a high-finesse femtosecond enhancement cavity with an on-axis obstacle. By inserting a wire with a width of 5% of the fundamental mode diameter, the finesse of F = 3400 is only slightly reduced to F = 3000. The low loss is due to the degeneracy of transverse modes, which allows for exciting a circulating field distribution avoiding the obstacle. We call this condition quasi-imaging. The concept could be used for output coupling of intracavity-generated higher-order harmonics through an on-axis opening in one of the cavity mirrors.