Most of the time, I believe (as most people here) that Boltzmann's constant is "just" an arbitrary unit conversion factor (temperature $\Leftrightarrow$ energy), and that we could get rid of it. It's not a fundamental constant imposed by nature.
However, when I'm thinking more about it, I'm frequently having a doubt. Here's my - unsure - opinion that $k_{\mathrm{B}}$ may actually be a very deep fundamental constant, like $\hbar$ and $c$ (which themselves aren't simple unit conversion factors, as I'll try to show below).
We define statistical entropy as this (of course, $p_n$ are probabilities, but I'll not elaborate on this):
\begin{equation}\tag{1}
S_{\text{stat}} = - k \sum_n p_n \ln{p_n},
\end{equation}
where $k$ is an arbitrary positive constant with any units. It would be simpler to just use $k \equiv 1$ by definition (without any unit), or to choose $k = 1/\ln{2}$ in units of "bits" to make a contact with Shannon's information theory. By its definition, (1) is not something that we can actually measure in a laboratory.
But then, when we want to give a relation with this quantity and the things we can measure in a laboratory, using macroscopic bodies, we have to introduce a coupling constant which states that:
Extracting information from a macroscopic system has an energy cost.
This cost is imposed by nature. This implies that the coupling constant between a measurable quantity (energy, for example) and the non-measurable statistical entropy (1) is not 0. Boltzmann's constant is a measure of that cost.
In this context, it's natural to set $k \equiv k_{\mathrm{B}}$, so the coupling constant is included in the definition of the statistical entropy (which could be identified with the thermodynamic entropy, i.e the one which appears in the empirical/physical relations). Its small value in human units ($k_{\mathrm{B}} \sim 10^{-23} \, \mathrm{J/K}$) is a manifestation that information is cheap in our universe. Humans can get a lot of information about Nature by doing measures which aren't removing much energy from us (the cost is low).
In principle, we could imagine an hypothetical universe in which extracting information is extremelly painfull. The cost is then very high and $k_{\mathrm{B}} \sim 10^{12} \, \mathrm{J/K}$ in this universe (using the same units as in our universe, which may not have any meaning actually!). We could define another universe in which the cost is infinite: $k_{\mathrm{B}} \rightarrow \infty$. Observers couldn't get any information at all by any measure in their lab. Life wouldn't be possible in that universe. On the opposite side, we could imagine another universe in which information is totally free: $k_{\mathrm{B}} \rightarrow 0$. In that case, any tiny measure could bring a lot of information and life would be easy (actually, way too easy. Life would probably destroy itself by overpopulation, since living organisms could be immortal!).
Once you get $k_{\mathrm{B}} \ne 0$, you could introduce a system of units which gives $k_{\mathrm{B}} = 1$. The "arbitrary" choice $k_{\mathrm{B}} \sim 10^{-23} \, \mathrm{J/K}$ is a way of stating our size (scale) in our universe.
I believe there's something similar with other constants of Nature which are usually interpreted as simple unit conversion factors, like $\hbar \sim 10^{- 34} \, \mathrm{J \cdot s}$ (unit of action) and $c^{-1} \sim 10^{- 9} \, \mathrm{s / m}$ (unit of time in spacetime).
The important point to notice isn't the particular (small) value of these constants (which depend on our scale in the universe), but the fact that they aren't 0 in our universe. Our universe isn't just a newtonian world, for which $k_{\mathrm{B}} = 0$, $\hbar = 0$ and $c^{-1} = 0$. From this point of view, the Boltzmann constant isn't just arbitrary: It's a fundamental property of our very large and complicated non-newtonian world, and that the living beings are defining a special scale, the only one at which Life is possible.
