With many CMOS op-amps, the output voltage will be zero, providing the output stage doesn't sink any current.
For an op-amp with a CMOS output stage, the saturation voltage at low currents can be predicted if you know the on resistance of the output MOSFETs. Unfortunately this isn't given on the datasheets. Out of the op-amps mentioned in this thread, the OPA340 is the only one which gives a clear graph showing the output saturation, when the MOSFETs are operating in the Ohmic region, when the voltage drop is directly proportional to the output current.
When operating off 5V, the N-MOSFET on the OPA340 has an on resistance of between 7.5

and 30

, depending on the temperature.
20 bits of accuracy with a full scale of 5V, is 4.77µV per count. Assuming a typical output resistance of 15R, the maximum current which the op-amp can sink and still remain under one count is 318nA, so the maximum resistor you can connect between the output and 5V is 15.73M.
However the OPA340 isn't suitable for this application, as the offset voltage is typically 150µV.