Is it a Munchausen Number?
05AB1E, 7 bytes

LвDmOQZ


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The larger test cases will time out on TIO.

Explanation

L         # push range [1 ... input]
 в        # convert input to a digit list in each of these bases
  Dm      # raise each digit to the power of itself
    O     # sum each
     Q    # check each for equality with input
      Z   # max