Maximum number of vertices after clipping a triangle against an AABB
Funnily enough, I asked this exact question on Math.SE a couple years ago: Maximum number of vertices in intersection of triangle with box.

The answer is 9 vertices, because each of the 6 planes of the box can cut off one corner of the polygon, replacing one vertex with two. So 3 vertices + 6 added vertices due to clipping = 9 total.