[amluto]

> For example, you can't define entropy as an absolute number; it needs to be assigned a unit,

You can define entropy directly without reference to other units, although it's a bit awkward. Entropy is the log of the number of microstates that correspond to a system's macrostate. Concretely, if you put n mols of ideal gas molecules in a box of volume V at a pressure P and temperature T, there is some large number of microstates corresponding to all those parameters. Entropy is the log of this number.

In classical mechanics, there's a normalization problem if you try to get an actual number out of this type of problem -- the microstates and all the macroscopic parameters are continuous. In quantum mechanics, though, this issue is solvable, although it's still awkward.

I can imaging a different type of system in which entropy really can be calculated, though. Imagine a particle that can be in exactly one of two states that are macroscopically identical. Now try to cool the system so that the particle is in one of those states of your choice. To do so, you will need to dump exactly 1 bit of entropy.

1 bit of entropy is tiny, but adiabatic demagnetization refrigerators work kind of like this, albeit in reverse, and I could imagine an experiment that would use a device like an adiabatic demagnetization fridge to remove a calibrated number of bits of entropy from some object. From this, you could, in principle, define entropy directly.

[URSpider94]

While you are right that this is the statistical definition of entropy, there are two issues: first, entropy is a unitless quantity, so defining it exactly doesn't actually move the ball forward in terms of defining units of measure. Second, outside of its theoretical underpinnings, physicists and chemists hardly ever talk about absolute values of entropy, they almost always use differences in entropy -- which factors out the need to define the exact number of microstates present before and after.

[amluto]

> first, entropy is a unitless quantity

The parent post asked about defining temperature in terms of more fundamental units. Entropy is typically written in J/K. If you treat it as dimensionless, then you get a definition of temperature in terms of energy for free. My point is that you can, in principle, actually do an experiment to make this useful.