Which Chandrasekhar Limit do I use? 1.39 or 1.44? Different sources online say that the Chandrasekhar Limit is either 1.40 or 1.39, or 1.44 solar masses. Why the discrepancy? I heard it might have to do with the composition of the white dwarf, but, correct me if I'm wrong, don't pretty much all white dwarf stars have the same composition?
The "classic" Chandrasekhar mass is given by
$$ M_{\rm Ch} = 1.445 \left(\frac{\mu_e}{2}\right)^{-2}\ M_{\odot},$$
where $\mu_e$ is the number of mass units per electron ($\mu_e =2$ for ionised carbon, oxygen or helium; $\mu_e = 1$ for hydrogen, $\mu_e= 56/26$ for iron (56)).

This assumes a white dwarf star of uniform, ionised composition and an equation of state given by that for ideal fermions (electrons in this case). It also ignores rotation and it uses Newtonian gravity. When defined in this way, the Chandrasekhar mass is obviously composition dependent - but most white dwarfs are expected to be formed from something with $\mu_e =2$.

In practice, a white dwarf can never become this massive. There are (at least) 4 effects that can mean the de facto Chandrasekhar mass (if it is taken to mean the maximum mass of a stable white dwarf) and at least one effect that can increase it.

The effects that reduce it:

Electrostatic interactions The electrons and ions do not form an ideal Fermi gas because of Coulomb interactions. The net result is to make a white dwarf slightly more compressible and the maximum mass about 2% lower. The correction is composition dependent; it is stronger for white dwarfs made of material with a larger atomic number.

General Relativity Massive white dwarfs are strongly affected by General Relativity. White dwarfs will become unstable at a finite density when the hydrostatic equilibrium is treated with GR. This finite density is reached at about $1.38-1.39 M_{\odot}$.

Inverse beta decay At high densities the electron Fermi energy becomes high enough to initiate inverse beta decay reactions. Electrons are captured by protons in the nuclei and this renders the star unstable. For a Carbon white dwarfs this will also happen at about $1.39 M_{\odot}$, for oxygen the threshold is lower at $\sim 1.37M_{\odot}$.

Pyconuclear reactions At high densities, even at low temperatures, fusion reactions can be initiated by quantum tunnelling. This changes the composition of the white dwarf and can change $\mu_e$ or lower the density threshold for inverse beta decay making the star unstable.

The effect that can increase the maximum mass for stability is rotation. Some authors have claimed that the limit can be increased to as high as $2.6M_{\odot}$ in certain circumstances, though these are usually referred to as Super-Chandrasekhar mass (e.g. Das &amp; Mukhopadhyay 2013) and are not stable. A stability analysis of rotating white dwarfs in GR found that the maximum stable mass was only increased to around $1.47 M_{\odot}$ for a Carbon white dwarf (Boshkayev et al. 2012), but these would be rotating faster than any observed white dwarfs.