One possibility: learn about the zoo of sequences.
If you have a rule for generating numbers: a first, second, third, fourth, and so son--what does the list look like? The order matters, so you can number the terms in the sequence if you need to: term 1, term 2, etc: T(1), T(2), T(3), etc. You usually need to number them.
The answer to that question has applications in measurement, chaos theory, control systems, and on and on.
Zeno knew about this one 2500 years ago: (1, 1/2, 1/4, 1/8, 1/16, 1/32 ...). The next term in the sequence is half the previous term. The numbers get small pretty quickly--closer and closer to 0 but never reaching it. This kind of sequence "converges:" it has a limit.
A boring example is (1, 1, 1, 1, ...).
You've probably heard of Fibonacci's sequence: (1, 1, 2, 3, 5, 8, 13, 21, 34, ...). The next term in the sequence is the sum of the previous two, and it gets larger without any limit ("diverging"). You can show that if you go farther and farther along, that the ratio of a term to the one before it is (1+√5)/2.
Some do neither: (1/2, 1/2, 1/4, 3/4, 1/8, 7/8, 1/16, 15/16, ...). The first, third, fifth, etc (odd) terms converge to 0, while the second, fourth, sixth,etc (even) terms converge to 1.
It isn't hard to predict what this will do: (1, -1/3, 1/9, -1/27, 1/81, ...)
Pick a rule. How about a continued fraction? Start with something something simple: T(n) = (1+1/(1+1/T(n-1)))
Let the first term be 1. The second is 1+1/(1+1/1) = 3/2. The third is 1+1/(1+1/(3/2)) = 8/5. Look at the Fibonacci sequence and see if you can guess that the fourth term of this sequence will be.
Suppose we use T(n) = 1 + 1/(2 + 1/T(n-1)). Start at 1. (1, 4/3, 15/11, 48/37, 173/133, ...). That converges to (1+√3)/2 or about 1.36602...
You can start at other numbers besides 1. How does the sequence change?
Things can get downright weird. Suppose you define a sequence T(1) =1, T(2) = 1, T(n) = T(n -T(n-1)) + T(n -T(n-2)). This is "Hofstader's Q Sequence." When a sequence gets a name, you know that there's either something very important or something very weird about it. The first numbers are pretty easy: (1, 1, 2, 3, 3, 4, 5, 5, 6, 6, 6, 8, 8, 8, 10, 9, 10, ...). As you keep on going, though, there can be wild swings. Mostly the numbers stay close to T(n) ≈ n/2, but they jump around a lot. These aren't random numbers--they're all well-defined. However, it isn't even known if the sequence dies somewhere along the line, by trying to refer to an undefined term (like the -1'th term).
The heaviest algebra I've referred to so far is the quadratic equation.
I'm not the first to think of different math directions; I got the idea from learning about some people who were trying to develop materials for exactly that purpose.
The New Mathematical Library series of books was "written by professional mathematicians in order to make some important mathematical ideas interesting and understandable to a large audience of high school students and laymen." The book from that series on continued fractions is available as a PDF.