A 3D shape with finite volume but infinite surface area. You can fill it with paint but never paint its surface.
From Wikipedia
Gabriel's horn is a type of geometric figure that has infinite surface area but finite volume. The name refers to the Christian idea that the archangel Gabriel will one day blow his horn to announce Judgment Day. The properties of this figure were first studied by Italian physicist and mathematician Evangelista Torricelli in the 17th century.