Confining photons in the smallest possible volume has long been an objective of the nanophotonics community. In this Letter, we propose and demonstrate a three-dimensional (3D) gap-plasmon antenna that enables extreme photon squeezing in a 3D fashion with a modal volume of 1.3 x 10(-7) lambda(3) (similar to 4 x 10 x 10 nm(3)) and an intensity enhancement of 400 000. A three-dimensionally tapered 4 nm air-gap is formed at the center of a complementary nanodiabolo structure by ion-milling 100 nm-thick gold film along all three dimensions using proximal milling techniques. From a 4 nm-gap antenna, a nonlinear second-harmonic signal more than 27 000-times stronger than that from a 100 nm-gap antenna is observed. In addition, scanning cathodoluminescence images confirm unambiguous photon confinement in a resolution-limited area 20 x 20 nm(2) on top of the nano gap.