I don't know if this really qualifies, but I would say that Scott Draves 
can be viewed as an amateur mathematician for inventing/discovering the fractals known as 
Flame fractals. His work is more towards art, but there is a decent amount of math behind
to optimize the aesthetically properties of the fractals. 
(The "nicest" fractal dimension is 1.52, for example).

Another more recent contribution regards [superpermutations][1], by G. Egan and an anonymous 4chan user. See also [this quanta magazine post][2].


  [1]: https://www.quantamagazine.org/unscrambling-the-hidden-secrets-of-superpermutations-20190116/
  [2]: https://www.quantamagazine.org/sci-fi-writer-greg-egan-and-anonymous-math-whiz-advance-permutation-problem-20181105/