I implemented a lot of statistics functions to learn how to program in a language. Some of the things I've learned on include:

  * moving averages
  * linear regression
  * matrix operations
  * matrix lu factorization
  * polynomial interpolation.
  * multiple linear regression
  * cubic spline interpolation
  * variance, covariance, correlation
  * gamma function
  * t-test, t-value calculations

One limitation of the above is that it doesn't explore all of a language's features. The above are mostly interested in array processing. There aren't any classes or generic types or what-have-you.