Live your DeepDream: how to recreate the Inceptionism effect

In the last few months the Internet has been flooded with deep dreams: images augmented by neural networks which look incredibly trippy. Deep dreams have the potential to become the new fractals; beautifully backgrounds everyone knows are related to Maths, but no one knows really how. What are deep dreams, how are they generated and what can they teach us? A neural network gets an image as an input, and returns a classification result: yes, that's a face. It achieves this by recognising features in an hierarchical fashion.