Chip-firing, rotor-routing and cycle-popping can be understood by anyone with or without combinatorial background, and provide new insights on lots of old combinatorial problems (counting Eulerian cycles and spanning trees, for instance). Here are some more basic facts, and here are some newer results. While the papers linked are probably too concise and too scholarly to be understood by students directly, it shouldn't be that difficult to make the results accessible for school students by writing them down in a more expository manner. (Needless to say, this would actually add a lot of value.) There are some notions from algebra used (group, group action, monoid, determinant), but (except for some linear algebra, which also can be avoided if so desired) mostly just the language is being used, not any nontrivial theorems.

Gröbner bases and elimination theory are another good field, but I don't have a good elementary reference for this. The question how to solve a system of polynomial equations in general is a natural one and a good student should have asked himself this question at least once. Unfortunately the answer is never given even in university lectures. Algebraic geometry is not an answer.

Now that we are talking about solving equations, I remember Vladimir Arnold having written a school-level (well, something he considered school level, referring to Russian schools) treatment of a topological proof (or an almost-proof, up to some intuitively obvious technicalities that should be cleared up in an analysis course) of the unsolvability of the generic quintic in radicals. Unfortunately I remember neither the proof nor the source, and it might be just my imagination...

EDIT: Here is the text (not by Arnold, but based on Arnold's lectures). It is much longer than what I had in my memory, although the price tag of over \$100 is questionable... You can get the Russian original for free, but then again with some rudimentary Russian you can just as well get the translated book in djvu...

PS. I got from chip-firing to Gröbner bases through a curious and tremendously useful mathematical fact, the Newman lemma (often also called diamond lemma by algebraists, whereas computer scientists use "diamond lemma" for a much easier version of this fact), which is (sometimes) used in proving the basic facts of both of these fields. While it can be avoided in both chip-firing and Gröbner bases, I think it should at least be mentioned (the proof is a wonderful exercise on algorithmic thinking) for the sake of general education.