That question is ambiguous. If you only need a single cat to go under the rope at some point, you just add a small arch for the cat to walk through. This requires a length of rope somewhat less than twice the height of the cat.

2 x Pi x Cat would levitate the rope by the height of the cat along the entire circumference of the Earth, allowing a billion cats to go under it simultaneously.

You could even make do with no additional rope if you're allowed to take advantage of terrain.