• Lagrange's four square theorem
• High-speed computer multiplication using the FFT
• The zeta function and the Riemann hypothesis (i.e. to understand what they are and why they're related to prime numbers; they're not expected to prove the RH)
• Descriptions of extremely big numbers (Conway's chained-arrow notation and its recursive definition, and stuff like that.)
• Proof of Fermat's last theorem for exponent 4 (I think this is supposed to be pretty accessible)
• Hilbert's tenth problem
• Basics of error-correcting codes
• Heuristic estimates for difficult or unsolved problems like FLT, Goldbach, etc. (see e.g. http://terrytao.wordpress.com/2012/09/18/the-probabilistic-heuristic-justification-of-the-abc-conjecture/ )
• combinatorial identities
• The formula $1+2+\cdots+n=n(n-1)/2$ is well known. What are the corresponding formulas for sum of squares, cubes, etc.? This may be closer to high-school level.
• the Ulam spiral

etc.