
And the answer is infinitely many.

If you thought it's 3, then yescubes might have 3 vanishing points,

like this one over here, which is a cube under perspective projection.

It actually has 3 vanishing points:

No. 1, No. 2, and No. 3 over here.

But that's because the cube has 3 faces.

You've had different faces whose Zdistance to the camera varied.

And those enclosing lines that might be parallel in the physical space

would result in their own vanishing point.

So you can theoretically make an object with infinitely many vanishing points.