The heaviest nuclei decay almost exclusively by spontaneous fission. The heavier they are, the shorter their half-lives. Any heavier nuclei than those now known will have even shorter half-lives.
Their have been some suggestions, based on the 'nuclear shell model', that certain "magic numbers" of protons and neutrons might lead to more stable nuclei. The concept is sufficiently well-established that I've seen such suggestions in chemistry textbooks I've taught from, and in some SF stories as well. But "more stable" -- at least to most researchers -- means somewhat more stable than their neighbors, which have very short half-lives -- but such heavy nuclei are difficult to model with current nuclear theories, so the vote is not unanimous, and some optimists think such elements might be isolable. Personally, I think it's unlikely that we're going to find new stable elements beyond those already known. It would just require too many questionable assumptions to all line up favorably for such a search to succeed.
Discussions of stable vs unstable nuclei need to include the 'Liquid Drop Model' as well as the shell model. The LDM predicts very well which nuclei will be stable to beta-decay, with only a few exceptions due to the effects of "magic number" shells. (It's actually an interesting exercise to step through successive atomic mass numbers, A, and predict, for each one, what the atomic number(s) of the most stable nuclei will be (one for odd A, two for even A). The LDM alone usually leads to the correct prediction. Exceptions are instructive, as they always involve cases where the optimum balance between protons and neutrons is very close to a tipping point -- then, shell effects decide. Away from such tipping points, the magic numbers don't make much difference.) All sufficiently heavy nuclei spontaneously fission to form more stable nuclei; the LDM correctly predicts these decays will be more energetically "downhill" the larger the nucleus, and the trend is continuously downward as the mass number increases.
In principle, anything's possible, but some things are just much less likely than others.
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