Solution below.

Reveal solution by highlighting:

500!*499!*…*2!*1! can be written in another way. 500!*499! is also 500*(499!)², so it’s 500 and a square. We can do this all the way through and get 500*498*…*4*2*(a bunch of squares).

This can again be written another way. 500*498*…*4*2 = 250*2*249*2*…2*2*1*2 = 250*249*…*2*1*2²⁵⁰. That’s 250! and a square.

So if n = 250, we would for the whole expression have a bunch of squares and nothing else. Anything smaller wouldn’t work, because there would be 250*249*… left over, with no neat behaviour. So n = 250.