Non-traditional math concepts for early education
I've had success teaching an elementary school student about binary numbers. I listed powers of two up to 512 and claimed I could make any number up to 1000 by adding the powers of two, using each one at most once. I had the student give me numbers and demonstrated how to make them out of the powers of two.
Then I had the student do it herself for some small numbers using guess-and-check. Then we made a table with powers of two as columns and sequential integers, from 0, as rows, and put an X in any entry if the power of two for that column was used to make the number in that row.
We looked for patterns in the X's, and were introduced to binary numbers that way. By the end of two hours, she could convert any number between binary and base ten, and could add and multiply in binary.
I haven't tried to do it and don't know how, but I suspect you can introduce basic group theory to students of this age, too.
Prologue
I am $13$ years old, which is neither advanced nor elementary. In this post, I'd recall the mathematical concepts that fascinated me a lot in the past years.
A good thing to learn about is elementary number theory. I learnt how odd numbers are in the form $2k +1$ and even numbers are in the form $2k$, where $k$ is another integer. It was really interesting to tackle the proofs related to this. The realization came to me that remainder is not just another number when I studied modular arithmetic. It was a whole new experience. One is never taught elementary number theory throughout the elementary school (and mostly high school) years!
Besides that, I learnt about number bases and how to convert them. I also watched the TED talk the OP shared succeeded by buying Wolfram Mathematica. It's a great software and I use it for manipulations I couldn't do by hand. Also, I used my resources a lot: MIT OCW, Khan Academy, OpenStudy et al.
Teaching an elementary MathJax is enough for him to go to math forums and ask (So did I!). A little remark from my side: It'd be really, really good to have a site like Math StackExchange for juniors, where a little leniency is shown to newbies.
Epilogue
The above post is self-centered (sorry!), but the things I mentioned are good enough for any other elementary.