Surface phonon polaritons are electromagnetic waves that propagate along the interfaces of polar dielectrics and exhibit a large local-field enhancement near the interfaces at infrared frequencies. Theoretical calculations show that such surface waves can lead to breakdown of the Planck's blackbody radiation law in the near field. Here, we experimentally demonstrate that surface phonon polaritons dramatically enhance energy transfer between two surfaces at small gaps by measuring radiation heat transfer between a microsphere and a flat surface down to 30 nm separation. The corresponding heat transfer coefficients at nanoscale gaps are 3 orders of magnitude larger than that of the blackbody radiation limit. The high energy flux can be exploited to develop new radiative cooling and thermophotovoltaic technologies.