Disclaimer: photosynthesis is not my field of expertise.
I assume you are asking about the amount of light needed for photosynthesis to take place, not the light intensity needed to sustain the plant? Since photosynthesis is an interaction between chlorophyll and single photons I would assume that photosynthetic reactions could take place with just single photons of suitable wavelength (i.e. at the limit of defining light intensity), but to measure this might be extremely hard in practice. The article Photosynthesis in the Abyss indicates that this might be the case, saying that in deep sea environments "A single molecule of bacteriochlorophyll receives a single photon only about once every 8 hours" (see also this popular science article). The conditions for deep sea photosynthesis is described in Beatty et al. (2005), which report photosynthesis by green sulfur bacteria at hydrothermal vents, where the only source of light is geothermal radiation.
However, as stated earlier, photosynthesis is not my subject field so I do not know how credible this information is.