The electrons in a white dwarf are highly degenerate with $E_F/k_BT$ of several hundred, even at temperatures of tens of thousands of K. The only way to increase the Fermi energy is to add mass to the white dwarf. White dwarrfs of constant mass do not get significantly smaller or denser as they cool and so the Fermi energy does not significantly increase with time.

A typical white dwarf (of mass $0.6M_\odot$) has an average density of a few $10^{9}$ kg/m$^3$, is made of ionised carbon and has a typical electron Fermi (total) energy of 0.8 MeV, whatever its temperature. More massive white dwarfs have higher Fermi energies, less massive white dwarfs have lower Fermi energies. They are "of order MeV". Actually, the stuff you've written about the Chandrasekhar mass has nothing to do with the question at all. I've reversed my upvote for your needless edit.

@ProfRob Before becoming a white dwarf, a star is hot. Fuel exhausted, it goes through a path in the Hertzsprung-Russell diagram en.wikipedia.org/wiki/Hertzsprung%E2%80%93Russell_diagram but eventually it radiates away and shrinks. Once it is degenerate, only then does it deserves the name "white dwarf". But the process of cooling does happen, before it has this name. The fact that there is an upper limit to the Fermi energy is exactly what is the meaning of Chandrasekhar mass. Why else are there no white dwarfs with higher Fermi energy ?

The question is about white dwarfs and I don't understand your comment about maximum Fermi energies, as I said for a given WD it doesn;t change. The "Chandrasekhar mass" - as explained by you, is the mass at which the radius of the WD shrinks to zero and the Fermi energy does in fact become infnite. The actual upper limit, is set by General Relativity (not considered by Chandrasekhar) or by electron capture and both occur when the Fermi energy reaches around 10 MeV in the core of the white dwarf. None of which has anything to do with the OP's question. Your downvote has been reciprocated.

@ProfRob I never wrote that the Fermi energy varies for a given WD, but that it varies with the mass of the WD under consideration and that if, as you wrote yourself there is a maximum Fermi energy, it is because WD cannot have an arbitrarily large mass. What happens if one cannot stop at the WD level (stop at neutron star or all the way to BH) was well beyond the OP's question.

There is a maximum Fermi energy - it is arounfd 10 MeV (for a carbon white dwarf) as I said above. However, that is irrelevant to the question and I did not write it in my answer, so I'm not sure what you are on about. My downvote is because your answer and your comments give the impression that you do not understand that the Fermi energy of a white dwarf is more-or-less fixed as it cools.

@ProfRob You do mention a maximum mass of order MeV in your answer. Please reread it. You give no reason for that. I did, in mine. OK, that was not strictly needed to answer the OP's question. But I thought is was a nice addition. Where you read in my answer that I wrote that the Fermi energy of a given MD varies puzzles me. I was describing the cooling of a given star from exhaustion of fuel to WD state, depending on its mass.

I don't mention a maximum Fermi energy at all. I don't use the word "maximum", or the phrase "Fermi energy". What I discuss is that the electrons in a white dwarf have a distribution of energies from zero up to some large value. That is because the electrons do have a distribution of energies from zero up to the Fermi energy and the Fermi energy is "of order MeV", right?

Where you give your misleading impression that the Fermi energy can change: (i) The entire paragraph that begins "When the star is hot...". The star we are discussing in this question is a white dwarf. Even a hot white dwarf is highly degenerate througout 99 percent of its interior. The Fermi energy of the electrons in the bulk of the star will not change (significantly) as it cools.

If you meant that paragraph to be talking about some progenitor object then you need to be clear. However, the carbon core of an asymptotic giant branch star, that will become a white dwarf, is already highly degenerate.

(ii) "it is as if gravitational energy is unable to give more "degeneracy" energy to the electrons. But this is not so." You may understand what you mean, but this reads like you think that gravitational contraction can increase the Fermi energy.

(iii) "If it is too heavy, the energy reaches this value and keeps increasing. " ditto. The Fermi energy does not increase for a star of a given mass.

This is misleading. The degenerate white dwarf is already sitting at the core of its giant progenitor. The main reason it decreases in radius is losing its envelope. Yes, there is some shrinkage after this, but that is of the non-degenerate atmosphere which accounts for about 1 percent of the white dwarf mass and much less than 1 percent of its thermal energy.

I'm also really annoyed that you chose to directly criticise my answer, based on your own misreading of it, without having the courtesy of leaving a comment beneath my answer so that I know what you'd done.

Here in France it is late. I'd come back tomorrow (for me). Granted I should not have criticized your answer but I had the feeling that you were not adding much to mine and using "big words" while I was trying to explain the essential quantum nature of degeneracy with basic notions. And I did not know about the etiquette of leaving a comment beneath your post. More tomorrow.