Neural net that alters its own parameters
One can do whatever he likes.

If your algorithm allows richer structure modification than simply adjusting weights, then you can obtain a "net which alters itself", though this doesn't have any special/magical meaning.

Suppose for example that your net's inputs correspond to some properties of an image, and you have a " hidden node" whose input is the average pixel value. During training, you can see whether adding this node lowers your training error, and if so, add it. One can claim that the net magically realized that it needs to use the average pixel value, but this is merely a direct result of your learning algorithm definition.