Without the PEP solid matter would hardly be possible, let alone fancier structures like neutron stars. To quote a famous paper on the subject:
In the present paper we demonstrate that the effects [of removing the exclusion principle] would be even more drastic than those envisaged by Ehrenfest. We show that not only individual atoms but matter in bulk would collapse into a condensed high-density phase. The assembly of any two macroscopic objects would release energy comparable to that of an atomic bomb.
Ok, so we would be screwed if electrons didn't follow the PEP. However, there is an entire class of particles where the PEP does not apply - these are called bosons. You see, deep down what prevents two electrons from sharing the same state is that the electron has a spin of 1/2. An important theorem says that particles with half integer spin (called fermions) must follow a set of rules called Fermi-Dirac statistics. The PEP is the most important piece of this rulebook. In contrast to fermions, bosons have an integer spin, which allows them to follow a different set of rules called Bose-Einstein statistics. Now in the case of bosons you can cram as many particles you like in the exact same state. One key implication of this result is that as you reduce the temperature of a system of bosons towards 0K, they will fall into the lowest energy state possible for the system.
The fact that bosons follow different statistics is not just an arance fact, it has very real consequences. In fact, in certain cases we can even see them with our own eyes. For example, the fact that Helium-4 atoms have a net integer spin means that they can act as composite bosons. As a result, a blob of Helium-4 atoms will follow Bose-Einstein statistics, which at low temperatures allows them to form something called a superfluid. A superfluid is marked by the unique property that it can flow without any resistance (zero viscosity). This phenomenon also produces some seemingly crazy results, such as fluids that are actually able to move upwards and flow around a container.Source