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Over the next semester, I'll be teaching two classes of kids about mathematics that they won't be getting in school.
One class is approximately high-school level, and knows algebra; the other class is throughout primary and middle school, and doesn't know much algebra.

The trick about these classes is that they are driven primarily by the kids, with the teacher there only to introduce the topic and provide mild guidance. So the kids ultimately dictate where we end up, although I'll try my best to steer them in the right direction.

I'm thinking of having the older kids look at maybe building the reals out of the rationals. They'll be attempting to learn about elliptic curves from the other teacher, so we'll hope that we don't overload them.
Another possibility, either for them or the younger group, is the Cayley Klein geometries from a symmetry perspective.

Anyway, I just wonder if anyone else has other ideas? Things they wanted to learn about when they were that age? Most of my interests tend toward differential geometry and topology, so I can't dump that on the kids, so does anyone else have any ideas?

Hmm, well I remember when I was about ten my teacher had taught us everything from the geometrey curriculum so she got us to do research into pi, what it is and where it comes from.
Also when I was about 14 I did I study into zero and it's importance in society today.

I don't know if this helps though