Tinn (Tiny Neural Network) is a 200 line dependency free neural network library written in C99.
For a demo on how to learn hand written digits, get some training data:
wget http://archive.ics.uci.edu/ml/machine-learning-databases/semeion/semeion.data
And if you're on Linux / MacOS just build and run Tinn with the test file:
make; ./tinn
If you're on Windows it's:
mingw32-make & tinn.exe