A stable nonlinear optical point light source is investigated, based on field enhancement at individual, pointed gold nanocones with sub-wavelength dimensions. Exciting these cones with near-infrared, focused radially polarized femtosecond beams allows for tip-emission at the second harmonic wavelength (second harmonic generation, SHG) in the visible range. In fact, gold nanocones with ultra-sharp tips possess interesting nonlinear optical (NLO) properties for SHG and two-photon photoluminescence (TPPL) emission, due to the enhanced electric field confinement at the tip apex combined with centrosymmetry breaking. Using two complementary optical setups for bottom or top illumination a sharp tip SHG emission is discriminated from the broad TPPL background continuum. Moreover, comparing the experiments with theoretical calculations manifests that these NLO signatures originate either from the tip apex or the base edge of the nanocones, clearly depending on the cone size, the surrounding medium, and illumination conditions. Finally, it is demonstrated that the tip-emitted signal vanishes when switching from radial to azimuthal polarization.