jmc

The largest perfect square less than $555$ is $23^2=529$. Therefore, there are $23$ perfect squares less than $555$. The largest perfect cube less than $555$ is $8^3=512$. Therefore, there are $8$ perfect cubes less than $555$. However, we cannot simply add those two numbers together because there are numbers that are both a perfect cube and a perfect square. For a number to be both a perfect square and perfect cube, it needs to be a $2\cdot 3=6$th power. The largest 6th power less than $555$ is $2^6=64$, so there are $2$ 6th powers less than $555$. Therefore, there are $23+8-2=\boxed{29}$ integers that are either a perfect cube or perfect square.