Maximum Local Value for the Electron Density
Single electron systems

I believe that particularly high values of n[V] are unlikely because as you pack more and more electrons in the same region, their repulsion would start to overpower their attraction to the nuclei, meaning the electron density would likely spread out more.

The main constraint on high electron density is not the repulsion between electrons, but the Heisenberg uncertainty principle. Even a single electron has a limited average electron density, given its average momentum (see e.g. here). For single electron systems (hydrogen-like ions), you get the highest electron density by increasing the nuclear charge. According to the virial theorem, as the potential energy of the electron increases (the magnitude, so tighter binding), the kinetic energy also increases. This will go along with a higher momentum, allowing the electron to occupy a smaller volume (i.e. have a higher density).
Multi-electron systems
For multiple electrons, theory gets complicated. X-ray diffraction gives experimental data related to electron density, but you need extremely high resolution (and extremely low static and dynamic disorder) to be able to map the electron density near the nucleus. In this study on LiF and NaF, the electron density at the lithium nucleus is estimated as 50 electrons per cubic angstrom using maximum entropy methods (panel b).

Fluorine, with a higher atomic number, will have higher electron density at the nucleus, but the density is cut off in the figure so you can't see the maximum value.

I understand that such a value would vary depending on promixity to the atomic nuclei (which would shield electrons from their own mutual interactions)

The innermost electrons would not be shielded, so the maximum electron density for a uranium nucleus with one electron would not be much different from one with 92 electrons. In fact, the binding energy of the inner electrons was used to figure out atomic numbers using Moseley's law.

Is there a maximum value for n[V] above which the mutual repulsion of electrons would mean the density was unreal?

Electron repulsion is not the main constraint on high electron density. To get high electron density, you need a high nuclear charge. At some point, relativistic effects become important, and for super-heavy nuclei, electrons get captured.
There was a comment from Martin about the use of the word &quot;unreal&quot;. I agree with Martin that if you observe a certain high electron density, that is as real as it gets. You might run into a calculation claiming a ridiculously high electron density, and then might conclude that results are not plausible and warrant further research.
So where can I find high electron density?
That would be near the nucleus of the heaviest elements as long as they are stable at least for a while. Using the arguments made above using the uncertainty principle, the binding energy of inner electrons, and the associated momentum of the electron (or looking at the Bohr model), the radius $a_0$ should be proportional to the reciprocal of the atomic number, meaning the electron density increases with the cube of the atomic number.