After 93 years, testing a theory as successful as General Relativity might seem pointless, but physicists hope to someday see cracks in Einstein's armor. "More than ever before astronomical discoveries are driving the frontiers of elementary particle physics, and more than ever before our knowledge of the elementary particles is driving progress in understanding the universe and its contents." That's what a National Academies of Science panel said in the 2003 report, Connecting Quarks with the Cosmos: Eleven Science Questions for the New Century.
With the fundamental physics questions requiring ever-more powerful atom smashers to answer, the report suggested, natural cosmology labs like PSR JO737-3039A and B are the place scientists will have to look for new physics beyond Einstein.
Maybe some day they will find something. But for now, you should probably bet on the guy with the crazy haircut.
NOTE: Einstein published General Relativity in 1916, although he finished it in 1915.