Ask HN: What is the simplest but still useful NN example you know?

I find neural networks pretty interesting, but most useful examples I know of require too much training data to be useful when it comes to trying to understand the whole process. I used a DOS program over 20 years ago in school to train an NN to do a simple table lookup. That isn't very exciting of course, since it is pretty obvious that a simple array would do much better.

I am wondering, are there any small, self-contained NN examples that actually do something useful, while still being small enough to play with an implementation in my favourite programming language?