Writing a piece for laymen which happens to mention nucleosynthesis, I realise that I do not know the answer to the following question:
Are there stable isotopes which do not occur in nature?
One could make an equally striking point either way:
Yes, there are isotopes which would be stable if any atoms of them existed, and we can make them in the laboratory, but conditions in nature have never led to them being created.
No, the unholy mess of a supernova explosion is such that anything that could possibly exist ends up getting made, even if only in small quantities.
I mentioned ${^{180}}\mathrm{Ta}^\mathrm{m}$ in the heading because it is frequently described as "the rarest stable isotope". Even if this is true, "the rarest stable isotope" is ambiguous, because it could mean "the stable isotope with the smallest abundance $>0$" (thus: the rarest of those that exist) or "the stable isotope with the smallest abundance $\ge 0$".
Specific example
Note that deuterium is hard to make in the Big Bang because it is easily photodissociated and so cannot exist at high temperatures. In this case, the kinetic energy required to make D is low enough that there is a range of temperatures at which D can be created but not destroyed. But consider another (imaginary) stable isotope - call it X - which requires more kinetic energy for its creation and is also more unstable to photodissociation. X would not be creatable in a hot gas, because a gas hot enough to make it would break it; but would be creatable in a cold environment: for instance with particles accelerated at a target and no hot radiation around to destroy it afterwards.
This is not to assert that such an isotope exists - merely to clarify the question.
