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?
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.
Further to the answer of @Jonathan, the thing that distinguishes one chemical element from another is the number of protons in the nucleus, which in turn determines the number of orbital electrons in the uncharged atom.
But we already know the element that corresponds to any given number of protons between 1 and 112; that's the atomic number. And you can't have a fraction of a proton. The only room for possible new elements is on the end.
Another way to look at this question is to consider how elements are produced. The elements with larger atomic numbers (i.e.: 26 (iron) or so) on the periodic table are primarily produced during supernovae explosions. Based on a lot of findings in stellar physics and nuclear physics in the past half century, it's unlikely that a transfermionic element (an element with 92 or more protons) can be produced in that process. Further, these elements tend to decay with half lives measured in hours or minutes (or less), so even if they were produced in a supernova, they are long since gone.
As @Jonathan pointed out, there is some potential for such elements due to the so-called island of stability, but they are still likely to be highly unstable, with very short decay times.
A chemical element is defined by the number of protons it contains, this largely defines its chemical properties. Elements can, within certain bounds, have varying number of neutrons (elements with the same number of protons but a different number of neutrons are called isotopes). Number of neutrons can have a subtle effect on chemical properties and a more significant effect on stability ie rate of radioactive decay.
But the big chemical differences which define an element are determined by the proton number and a given element will only have a handful of isotopes within a fairy narrow range.
So, elements are classified by the periodic table which lists the elements in groups according to atomic number (number of protons). When the periodic table was first proposed there were a number of gaps between known elements (at this point the existence of protons was not known). These gaps have subsequently been filled so there is no space for new elements untill you get to high atomic numbers.
The periodic table is full in terms of what could be considered reasonably stable elements. There is no fundamental reason why you can't propose elements with ever increasing atomic numbers. However the trend so far is that with increasing atomic number elements become more and more unstable. They can be created in particle accelerators but only exist for a tiny amount of time and don't exist in nature in any way which you might consider being a 'real' material like iron or copper.
There have been various predictions of theoretical islands of stability but even then we are talking about very short lived elements.
So in terms of the way that we tend to understand the term there are no new elements to discover, as all reasonably stable possibilities are accounted for.
Having said that there could well be entirely new materials composed of known elements or indeed previously unknown states of matter.