Well, first there might be unknown elements yet to be discovered that you create artificially, and that might have existed in our universe in the past.

They would be in the "island of stability" in this diagram

Island of stability

They have even been given names. One of the more stable elements in the island could be Unbinilium, or perhaps flerovium 298. There other possibilities, as there are various combinations of numbers of neutrons and protons that may be especially stable.

These island of stability elements could have half lives of seconds, minutes, or (probably less likely) even longer, up to millions of years. Sometimes scientists speculate that they may have long enough half lives to occur naturally in tiny quantities.  Superheavy Element 117 Points to Fabled “Island of Stability” on Periodic Table.

ANTI MATTER ELEMENTS

The main hypothesis is that all the elements we have have anti matter counterparts. So, if that is right, it would be the same periodic table in mirror form.

But it will be a while before we can check this. Unless we discover naturally occurring anti matter such as distant antimatter galaxies. None have been found yet, though there have been searches for anti matter galaxies. In Search of Antimatter Galaxies

If there were any nearby antimatter galaxies, well - it would be a spectacular event if one of them collided with a matter galaxy, and you'd expect to see dramatic events also when stars ejected from an antimatter galaxy enters into a matter galaxy. Discussion here on stackexchange How would we tell antimatter galaxies apart? (Science fiction story about an interloper anti matter star in our galaxy by Larry Niven Flatlander (short story) )

Also been searches for antimatter atoms such as hydrogen and helium in cosmic rays. Get plenty of positrons in electron positron pairs. But I don't think any anti atoms yet.

The other thing you can do is to make them here on Earth. They have created anti hydrogen atoms. And actually can slow it down and trap the atoms so potentially we can study it. Antihydrogen

They've also made anti helium nuclei. Not easy to trap it, and look at it carefully.

Physicists create heaviest form of antimatter ever seen

Antimatter and the Anti-Periodic Table

So we have two atoms of the table, hydrogen and helium. Will be much harder to create anti lithium though. We are a long way from creating a complete anti matter periodic table. If there are some differences between matter and antimatter perhaps it might not have all the same elements?

DARK MATTER ATOMS

Most ideas for dark matter involve matter that interacts weakly with itself and with ordinary matter. But some ideas suggest the possibility of "dark atoms". See New Kind of Dark Matter Could Form 'Dark Atoms'

NEUTRON STAR MATERIAL

You could consider a neutron star as a gigantic atomic nucleus consisting almost entirely of neutrons. So then - that brings up the idea of the element of atomic number 0 or "neutronium". Then any stable collection of neutrons bound together would be an isotope of this element of atomic number zero.

An isolated neutron then would be Mononeutron It undergoes beta decay with half life of about 10 minutes. So could be treated as a short lived atom of atomic number zero - which doesn't need any electrons because there are no protons in the nucleus.

The dineutron has been observed but is extremely short lived. They are not bound, but could perhaps be treated as a very short lived "atom" of atomic number 0.

The Tetraneutron  - which would be a bound state of four neutrons - was observed experimentally, (reported in 2001 Detection of neutron clusters) but their experiment has not been replicated. If it existed it would need revision of our understanding of nuclear forces.

Robert Forward in his novel Dragon's Egg suggests the idea of a chemistry in the material on the surface of a neutron star - so not the actual neutronium itself - but highly compressed white dwarf star material, basically ordinary matter - but with the atomic nuclei interacting directly with each other, under the strong nuclear force. So compounds are different, bound together by the strong nuclear force, but the nuclei are the same as for our chemistry.