I couldn't find anything specifically written about knowing the constants, and it doesn't seem any of the prep tests expect you to know them either.
Nonetheless, I would know Avogadro's number (6 x 10^23 molecules per mole), and the ideal gas constant (0.082 L*atm/(mol*K)).
I would also know STP (0 deg C = 273.15K, 1 atm) and standard conditions (25 deg C = 298.15K, 1 atm). You should also know things like the boiling and freezing points of water (100 deg C, 0 deg C, respectively).
Also, it might be a good idea to have some knowledge of molar mass of a few elements (H, O, N, C, etc.).
Beyond these things, I can't think of any numbers you may want to know but it can only help, IMO. Know things like the mass on e- is to the order of 10^-31, and the speed of light is 3x10^8 m/s....etc.