Temperature is a measure of the average kinetic energy per particle in an ideal gas (Kelvins = 2/3*Joules/kb) so the J/K units of kb is fundamentally unitless (Joules/Joules). kb is the conversion factor from energy in 3/2*Kelvins to Joules for an ideal gas. If kinetic energy measurements per particle of an ideal gas were expressed as Joules instead of Kelvins, kb in the above equations would be replaced by 3/2. This shows that "S" is a true statistical measure of microstates that does not have a fundamental physical unit other than the units of information, in this case "nats", which is just a statement of which logarithm base was chosen by convention.