Well I have no idea how things work in Blender, so I cannot help you there, sorry!

This may help:
Lawlor, Orion. (2012). GPU-Accelerated Rendering of Unbounded Nonlinear Iterated Function System Fixed Points. ISRN Computer Graphics. 2012. 10.5402/2012/825782.
  

So, blender's procedural texture nodes let you make loops? How?

There are no loops implemented but if you implement a function/transformation of initial uv coordinates (used mainly to align textures, normal maps etc) in math nodes than you can simply copy it n times. In my case i just dont know how to make setup for mobius transforms that converges to anything. I get some results by applying modulus before or after the transformation but i get discontinuity along the fractals. I know that there has to be some way of transforming simple xy coordinates to crteate kleinian fractal. My research will not stop untill i get it. Im even ready to pay a good money just to have setup for double cusp.

or use cube-map textures for the Riemann sphere...

Im not sure exactly what You mean. Blender does not support cube maps as far as im aware. It would not change anything in my case as i can simply take input called "reflection" instead of uv coord and i get similar effect to a cubemap but it still won't allow me to iterate mobius to create any limit set

This may help:
Lawlor, Orion. (2012). GPU-Accelerated Rendering of Unbounded Nonlinear Iterated Function System Fixed Points. ISRN Computer Graphics. 2012. 10.5402/2012/825782.
  

I will definitely take a look with hopes to understand this But im a simple man and if there is no xy formula - i will fail xD