Research over the past two decades has identified the weakest failure detectors for several important problems in fault-tolerant distributed computing. A recent work has shown that, for a certain definition of the term “problem,” every problem that is solvable using failure detectors has a weakest failure detector. In sharp contrast to these results, we prove that a fundamental problem in concurrent computing—FCFS Mutual Exclusion—is solvable using failure detectors, but has no weakest failure detector in the shared memory model. To the best of our knowledge, this is the first problem that is proved not to have a weakest failure detector. We also show that, if the FCFS requirement is dropped, the mutual exclusion problem has a weakest failure detector. In fact, we present the weakest failure detector for the more general problem of starvation-free k-exclusion, for any k.