Do elementary particles have a density? The SM supposes elementary particles are structureless unless composite objects like hadrons. For bosons, that can occupy the same state, we can define energy or mass density. The same happens but limited by Pauli principle. So, are energy or mass density for ensembles of bosons and fermions the only meaningful density for fields? Or are there reasons in the SM to consider electrons, fermions, even bosons as particles with size?The only size zero no infinite density particle I can oversee are massless particles. Thus, why stuff like classical electron radius are naive estimates of quantum sizes just as infinite density seems wrong? Is density energy quantum?And the density of a quantum or point particle?
 A: The SM defines elementary particles as point like, with no spatial extent and no substructure.
You are saying that you can only think of massless particles (like the photon, gluon, graviton) as defined by the SM (point like) that have no infinite density.
Why do you think that fermions, that do have rest mass are different? You are basically saying that particles with rest mass must have infinite density if they are defined by the SM as point like.
Now you are saying that in this context (having infinite density if point like), massless vs having rest mass makes a distinction.
I must disagree. Energy and mass are the same, you can transform them. What does it mean that a particle does have rest mass? Because of the mass-energy equivalence principle, in your case, for energy density, it does not matter whether a particle does or does not have rest mass.
Now you are saying that it is OK to have a photon point like, and have finite density. But it is not OK to have an electron point like, and have finite density.
I disagree with the distinction because:


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*both particles do have stress-energy, photons have frequency, electrons have rest mass and kinetic energy

*nobody has ever measured an electron at rest, rest mass is a theoretical value, calculated, and fits the theory of SM
Now basically, I believe that both massless and massive particles (elementary) are point like in the theory of SM. They both do have energy, and if you want the point like definition, then yes, their energy density is defined infinite. But to define energy density to a point particle is not possible, because of the HUP. You cannot restrict the particle to a volume of space after a certain size, without raising its energy (momentum) to infinity.
You can try to define these particles as point-like, but in reality they exist in spacetime, and QM fields exist throughout space, and these particles are excitation in those fields, which are not point like. That is the reason we distinguish near and far field. I will use photons for the example of massless and electrons for massive (gluons, quarks are in confinement, gravitons are hypothetical).
Now both photons and electrons do have energy, and create their own near field:


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*photons have stress-energy and create their own gravitational field

*electrons do have their own EM field and their own gravitational field 
Basically all elementary particles have their own force field. We use virtual particles to describe the math when we try to describe the interaction of these fields with other particles. So basically the size of these particles in reality is not point like.
If you like to use that view, then they do not have infinite energy density.
I do believe that maybe some day, when we will have figured out what elementary particles are made of (strings), then we will be able to understand energy density of elementary particles too.
