Beryllium-8 is an isotope of Beryllium, with four protons and four neutrons. Beryllium-8 has a half life of somewhere around 10 attoseconds,meaning it is ridiculously shortlived, even by the standards of radioactive materials. For this reason, Beryllium-8 could almost be described as too unstable to really exist, and most of what I am going to talk about is why it is conspicuous by its absence.
For most lighter elements, from hydrogen through calcium, having an equal number of protons and neutrons is a sign of at least relative stability, especially for elements with an even number of protons. Since Beryllium-8 fits this description, its extreme instability is somewhat curious. As far as I can understand, and I can't really claim to understand the science behind it, Beryllium-8 is so unstable because it is equal to two Helium-4 nuclei, which are so stable and self-contained that they naturally divide back apart. And it is Beryllium-8's role as two alpha particles that brings us to the next major role it doesn't play.
Inside of main sequence stars, the basic way that energy is produced is by fusing hydrogen into helium. Even in a relatively large star like our sun, this isn't very easy, because the temperatures and pressures in the center of our sun are barely able to counteract the electrical repulsion of protons. The sun has managed to keep burning for billions of years because it burns its fuel rather slowly. Most stars are red dwarfs, which are smaller and dimmer than our sun, and fusion proceeds even slower. So, at the temperatures and pressures inside of a normal star, once hydrogen has fused into helium, the next step in stellar fusion, which means both keeping the star burning and producing all of the elements heavier than helium, is to fuse that helium into Beryllium-8. But due to the aforementioned 10 attosecond halflife, the Beryllium-8 splits right back in two, and the entire process is foiled.
What does happen, is that in a star of sufficient mass and pressure, three helium nuclei collide at the same time, or two collide to form Beryllium-8, and in its attoseconds of existence, a third crashes in as well to form Carbon-12, which is also stable. From this, later generations of stars are seeded with Carbon-12 and heavier elements, that help them burn hydrogen, and also brings much richness to a cosmos that would otherwise be hydrogen and helium. But as it is, the instability of Beryllium-8 is the bottleneck, stopping the process of fusion in all but the heaviest of stars.
If we could somehow change the universe to be just a bit more like we could expect, and where Beryllium-8 was a stable isotope, the universe might look quite a bit different. Even smaller stars could produce a good amount of Beryllium-8 and thus heavier elements, and the interstellar medium would be that much richer. Although it is only the fourth element, it is solid at room temperature, so the amount of rocky planets in the universe would be much higher, and because Beryllium-8 would make the formation of Carbon-12 so much easier, carbon and complex organic molecules would also be much more common. In other words, if Beryllium-8 was stable, the universe might be a place where what we think of as "terrestial life" was much more common. In many ways, the strange instability of Beryllium-8 could be seen as an argument against intelligent design, since one of the most basic ways the universe could build up a more complicated structure is inexplicably blocked. However, as is often the case with intelligent design, it could just as well be argued that the seemingly miraculous triple-collision that overcomes the Beryllium-8 bottleneck is evidence that there may be some purpose on a deeper level. In any case, it is amazing the small details that go into producing the cosmos we live in. The strange instability of a form of an element we encounter very rarely is one of the keys to the reason the universe looks the way it does.