I really like the Collatz conjecture graph ($3n+1$ problem), the graph showing the evolution of integers under the rule:  If $x$ is even, divide by 2; if $x$ is odd, multiply by 3 and add 1.  Here's the graph with $100,000$ nodes:

[![Collatz graph][1]][1]

  [1]: https://i.sstatic.net/aZsUz.jpg