If the exclusion principle didn't hold, and all the electrons of one atom could occupy the same ground energy shell, then the atom would be inert and would not combine with other atoms to form molecules. No molecules means no life. The very existence of chemical bonds and biological structures is therefore explained by the fact that the exclusion principle does hold.
The exclusion principle is responsible for almost all the peculiarities of material world. It plays a fundamental role in solid state physics (electrical conductivity, paramagnetism, etc.) as well as in nuclear and subnuclear processes. Its effects are observable at any scale. For example, it enters in the collapse of stars to the white dwarf stage and in the further collapse to the neutron star stage.
As Pauli has written: ‘ The question as to why all electrons for an atom in its ground state were not bound in the innermost shell had already been emphasized by Bohr as a fundamental problem in his earlier works. […]However, no convincing explanation of this phenomenon could be given on the basis of classical mechanics. It made a strong impression on me that Bohr at that time and in later discussions was looking for a general explanation.' The exclusion principle answers this problem providing a simple and elegant explanation for the regularities of Mendeleev's periodic table of elements (by stating that no more than two electrons with opposite spin can stay in the same atomic shell.)