Which particular piece of hardware has crossed the Karman line most often?
By hardware I mean anything other than organisms.
The Space Shuttle Discovery has been to orbit a total number of 39 times, more than any other shuttle vehicle (source: Wikipedia). Contrasting that to other candidate objects:
Space Shuttles Atlantis, Columbia, Endeavour and Challenger flew 33, 28, 25 and 10 missions, respectively.
Spacelab pressurized module LM1 flew a total of 10 times.