Yes, there is a reason. We've thoroughly explored the periodic table through our studies of nuclear processes (lots of government funding for anything nuclear weapons related). It turns out you can create a ton of elements we don't see in nature, but almost none of them are stable: you smash atoms together to make a new one, but the new one flys apart after a <second (and that's actually a long lifetime for these atoms, many have lifetimes of nanoseconds). There is a potential "island of stability" around element 120-130, but even there lifetimes are predicted to be less than a minute.

In addition to the stability argument, there are also energy requirements. It turns out fusion (stars) only gives energy up to iron, then it requires energy to make bigger atoms. In other words, you get energy back out when you split big atoms (I.e. nuclear reactors). In nature, all elements past iron are created only in the spectacular energies of supernovas, which occur in less than a second.

Basically, we have a really good grasp on the elements that can exist. We're only missing the details on a few things that occur on nanosecond and less timescales.

The structure of elements/atoms is well understood based on their subatomic constituents. Naively, you might think that can you just keep combining increasingly larger numbers of electrons, protons, and neutrons to create new elements. However, the stability of an atom becomes problematic when the size of the nucleus approaches the interaction length of the strong force (i.e. the nucleus is too large for the strong force to hold it together). These elements are unstable and therefor not relevant as far as organic chemistry is concerned.

Furthermore, the formation of elements in the Universe is also a fairly well understood process. For elements lighter than Fe it generally occurs through nuclear fusion in the center of stars. For elements larger than Fe it generally occurs through the r-process and s-process. With these we can model nucleosynthesis extremely well and it gives us a very good idea of the elemental composition of the Universe. That being said, there could be some crazy unknown element out there but it would contradict almost everything know about atomic physics.

> Is there a reason that other (alien) elements can't exist that we've never been exposed to?

Yes, because you form new chemical elements by adding protons (and stabilizing neutrons) to the nucleus, and humans have found or synthesized the first 118 of these. The ones that don't occur in naturally on earth are short-lived an unstable.

If there was a lot of exotic elements out there, we'd have seen it with our telescopes. It would point to new physics, for a start, since we don't think very heavy elements are likely to form naturally at all, let alone be stable long enough to be observed in any quantity.

Even if such elements are out there in the universe in tiny quantities, we'd run into a billion carbon-based biospheres before we found any.

Physics is the reason that other elements likely don't exist. One can only arrange protons, neutrons, and electrons in a fixed number of "stable" ways.

Of course, this is today's understanding. We may be wrong. :-)