We realize giant optical nonlinearity of a single plasmonic nanostructure which we call a split hole resonator (SHR). The SHR is the marriage of two basic elements of nanoplasmonics, a nanohole and a nanorod. A peak field intensity in the SHR occurs at the single tip of the nanorod inside the nanohole. The peak field is much stronger than those of the nanorod and nanohole, because the SHR field involves contributions from the following two field-enhancement mechanisms: (1) the excitation of surface plasmon resonances and (2) the lightning-rod effect. Here, we demonstrate the use of the SHR as a highly efficient nonlinear optical element for: (i) the generation of the third harmonic from a single SHR; (ii) the excitation of intense multiphoton luminescence from a single SHR.