If n >= 10 we can show that n! + 5 is never a cube. This is because n! will be divisible by 100, so n! + 5 will be 5 mod 100. Therefore k^3 must also be 5 mod 100, which can never be true (since k^3 will have only one factor of 5, and it needs 0 or 3).
Therefore n must be less than or equal to 9. A quick check shows that n = 5 is the only possible n.
