What's the likelihood of the existence of unknown elements in the Solar System?
What's the chance that there might be undiscovered chemical elements in the Solar System - either on planets or around the Sun or on asteroids of the Oort-cloud?
If you mean elements with atomic number higher that 115-ish then of course there is a chance, so far nothing prohibits atoms from having as many protons - though stability is an issue. But I don't see the point in asking 'What is the chance..?'
Likelihood is "maybe": Superheavy Element 117 Points to Fabled “Island of Stability” on Periodic Table http://www.scientificamerican.com/article/superheavy-element-117-island-of-stability/
As far as elements (e.g. on the periodic table) go, I would say the odds are very slim. We already discovered or produced all the elements of the Periodic Table up to atomic number 112 at least. As the number increases, the half lives of the elements generally decreases, and is very short for elements above 102. If this trend holds true as the number increases, practically all the "undiscovered" elements should have turned into the lower known atomic number elements.
However, there is hope. There is a theorized "island of stability" where a narrow range of yet to be discovered high atomic number elements may be stable: http://en.wikipedia.org/wiki/Island_of_stability
I would say there is a slight chance this element could be discovered in the solar system.
Elements in the "island of stability" are expected to be *relatively* stable, compared to their neighbors. Quoting the Wikipedia article, "Specifically, they are expected to have radioactive decay half-lives of minutes or days, with "some optimists" expecting half-lives of millions of years." Even with half lives in the millions of years, there could still have gone through hundreds or thousands of half-lives over the history of the Solar System. Unless the optimists are underestimating their stability, there should be practically nothing left of them.