Can relatively simple gold structures, of nominal thicknesses 3-12 nm, provide significant electric field enhancement for use in cheap optical sensors? The nanostructures are fabricated using electron-beam deposition at 2Å/s onto room-temperature glass substrates and characterized by scanning electron microscopy (SEM), transmission electron microscopy (TEM), linear optical spectroscopy, two-photon photoluminescence (TPL), and Raman scanning optical microscopy. We demonstrate that the 6-nm-thin film, being close to the percolation threshold, at which individual gold nanostructures start forming continues layer, facilitates the highest local electric field enhancement. For some thicknesses, the TPL measurements even reveal super-cubic dependences on the incident power rather than the expected (and usually observed) quadratic dependence. We ascribe this feature to the occurrence of very strongly localized and enhanced electromagnetic fields due to multiple light scattering in random nanostructures that might eventually lead to white-light generation involving various high-order nonlinear processes.