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.
For example, you can contract a loop in the outside of a ball, but not in the outside of the horned sphere if the loop goes around the handle. However, with this approximation of the horned sphere, you can contract the loop because it branches only finitely many times. So here is a puzzle game that will test your patience and give you an intuitive understanding why it is impossible to contract the loop in the ideal case: Tie a thread around the big handle and try to free it by maneuvering it through the branches.
If this product is too expensive for you, try the less fat version, also available in my shop.