Seventh
Although it didn't make it to the list of the six functions in the Motivation part, I just couldn't leave this one out:
\(f(z) = \cos\,\dfrac{1}{z},\quad z \in \mathbb{C}(0,1)\)
The neighbourhood of zero with radius 1 as seen by the 'terrible Cosine Oneoverex'… Very exciting stories are related to this picture.