Reduce a number by dividing it by its number of prime factors, counting multiplicities.
Repeat this on the result as much as you want.
Is there an infinite number of squares that can be reduced to their roots?
4 and 16 reduce to their root in one step
1600 reduces to 40 in 2 steps
Bonus points if the question also gets answered for higher powers.
I did not brute force it, but would not be surprised if 16 reduces to 2 in 2 steps is the only one.