Is there anything in the universe that cannot be compressed?

Under special relativity nothing can be incompressible: consider any object of nonzero size and finite mass in its rest frame; when you apply a force to it on one side it will start moving. If it were completely incompressible, the other end would start moving simultaneously. Since the ends are spatially separated, there is a frame in which the other end would start to move before the application of the force, which is in contradiction with special relativity.

Elementary particles are generally assumed to be point-shaped, hence of zero size, so they would be trivially exempt.

In quantum field theory, an elementary particle doesn't have one precise location and size in space. The quantum of an electron field in free space has different extent compared to the electron around a hydrogen atom, for example (i.e. it's harder to bounce an electron off a free electron than off a hydrogen atom). While in one very real way, an electron's size has been probed up to the limits of our measurement ability and found to be extremely small and to our measurements indistinguishable from point size, in another very real way, the electron around a hydrogen atom is essentially as large as the atom.

To really get to the bottom of this, you would need to ask your friend: "How do you measure the volume of an electron?" Depending on the way you measure, you can get the result that either electrons are incompressible (within the limits of our measurement ability) or that they are extremely compressible. For bonus points, you can ask something like: "Are sound waves compressible? How do you measure the 'size' of a sound wave?"

It's true that there's a limit to how many electrons you can fit in a given volume of space. But that doesn't tell you much about the compressibility of the electron - just that it has some limit.

But your friend probably didn't have something complicated like that in mind. He's probably considering the classical elementary particle, which is point-size by definition. However, you need to keep in mind that this description is wrong - QFT is a theory that describes elementary particles much more accurately than the classical theory. All our physical theories work up to certain limits, and are either outright wrong or at least unconfirmed outside of these limits. Your question is outside the limits of classical theory; you're asking what happens when one impossibly small thing collides with another impossibly small thing.

But real electrons don't exactly bounce of each other the way rubber balls do - they're interacting through their respective electric fields, and even in the classical models, they never really get close enough to "touch" in the mechanical sense; the disturbances in those electric fields are what is pushing against the two electrons, and the closer they get to each other, the higher the force acting between them. In the classical model, where you have point-like electrons and no minimal possible distance, no matter how fast you throw the electrons against each other, eventually the force becomes high enough to accelerate the electrons away from each other fast enough to avoid the "touch"; they'll never occupy the same spot in spacetime, and they'll never be so close to each other that you couldn't fit another electron between them. Again, mind that this is not what actually happens; you're just using a model that doesn't give you good answers for the question you're asking.

When you start compressing ordinary matter, you first start by decreasing the space between atoms (after you have, almost mechanically, broken the bonds between molecules). This gets increasingly harder because the atoms are bouncing around and they are repelling each other, because when two atoms get close enough to each other, their electron clouds see each other and, since they are both negative, repel each other.

If you keep on compressing, this force gets bigger and bigger. At some point you are supplying enough energy to strip all atoms of their electrons and create a plasma, where the nucleii and electrons move independently, which gives you some more space to work with (i.e. compress). But at some point you still encounter a problem. The electrons still repel each other and zip around in the plasma. You cannot get them all to stand still at the same time because then they would be all in the same state, which is forbidden for fermions. This is actually what keeps white dwarfs from collapsing in on themselves.

But you know what wouldn't at least have this repelling? A bunch of neutrons. At some point the pressure you are exerting is enough to transform the electrons and protons into neutrons and neutrinos. The latter probably escape even from your magic compression device, but the former can now be compressed even more, since they are not electrically charged anymore. But neutrons are still fermions, so you run into the same problem again at some point, not all neutrons can occupy the same state at the same time. This is what keeps a neutron star from collapsing.

At some point you have compressed the (formerly known as) matter far enough that the volume lies within it's own Schwartzschild radius, and it collapses into a black hole. At this point General Relativity, currently our best description of gravity, predicts that your matter collapses into a singularity. Since a singularity has infinite density, it is arguably not compressible anymore and we are done.

However, this depends on how wrong you think General Relativity is. While it is the best we've got, we do know it must be wrong somehow, because it is not reconcilable with quantum mechanics (so far). Many think that a correct theory of quantum gravity would avoid a singularity, so you might be able to compress a black hole further. Or not, we do not know.