Conducting probe atomic force microscopy (CPAFM) was used to measure the electrical transport characteristics of 2-14 nm thick doped crystallites of the organic semiconductor sexithiophene (6T) grown on Au and SiO2 substrates by vacuum sublimation. To make the measurements, an AFM was modified to allow in situ switching from tapping mode imaging to point contact electrical characterization with an Au-coated tip. The crystals were characterized structurally by molecular contrast AFM imaging and consist of layers of 6T molecules oriented with their long axes nearly perpendicular to the substrate. For crystals grown on Au substrates, transport is probed through the thickness of the crystals (i.e., the vertical direction) using a CPAFM tip and the substrate as electrical contacts. On SiO2 substrates, transport is measured parallel to the substrate between the CPAFM tip and a nanofabricated Au electrode in contact with the crystallite. The measurements on Au reveal an unexpected dependence of the conductance on crystallite thickness, namely that conductance is greatest for crystals that are three 6T layers thick, not one layer. Both the vertical and horizontal conductance measurements show nonohmic behavior which may arise from an energy barrier to charge injection at the Au-6T interface. The reproducibility of the CPAFM methodology for probing transport in these extremely thin organic crystals and the observation of nonohmic behavior underscore the importance of nanoscale transport measurements afforded by CPAFM.