The 320 EeV air shower detected by the Fly's Eye poses an important problem. Careful analysis of pathlength limitations for the possible particle types due to cosmic background radiation verifies that the particle very likely traveled less than 50 Mpc from its source. The best candidates for accelerating particles to such high energies are the very powerful radiogalaxies, however they are all more than 100 Mpc distant. Our search finds no likely source within 50 Mpc in the direction from which the particle arrived. This prompts consideration of less likely astrophysical sources, like M82, as well as non-standard mechanisms like cosmic string annihilation. It is also conceivable that the air shower was produced by some non-standard particle whose pathlength is unlimited because it does not interact with the cosmic background radiation. A less radical alternative is that relatively strong magnetic fields deflected the particle's path through a large angle, so it could have originated at a nearby radiogalaxy at an earlier time of strong activity.