The Alexander horned sphere
is a topological shape whose interior is equivalent to the inside of a ball, but the outside is not equivalent to the outside of a ball.
This upper bound approximation of the horned sphere branches only finitely many times. At the lowest level, the branches are joined. So you can't use it to play the same game as with the open-ended version, also avaliable in my shop.
If this product is too expensive for you, try the less fat version, also available in my shop.