The M1+- surface of Wayne Rossman, Ed Thayer and Meinhard Wohlgemuth from 2000 combines the added handle from Fusheng Wei’s genus 2 surface with the handle from their own M1+ surface to make a doubly periodic genus 3 surface with 4 parallel annular ends.
This surface comes in a 1-parameter family. Numerically, one limit appears to be an 8-ended Karcher-Scherk surface with handle:
The other limit is more complicated. It appears that two doubly periodic Scherk surfaces with one tilted end each have been glued together end-to-end, with the intersections resolved by singly periodic Scherk surfaces:
One of the reasons to study examples of modestly high genus is that they suggest what new gluing constructions might be possible. The limits will often still consist of surfaces with explicit Enneper-Weierstraß data, making them amenable to direct computation and manipulation. Being able to perform gluings with elementary blocks like the Scherk surfaces (with tilted ends when needed) would cover a large class examples and likely also establish the existence of these surfaces as 3-parameter families.
Resources
W. Rossman, E.C. Thayer, M. Wohlgemuth: Embedded, Doubly Periodic Minimal Surfaces, Experimental Mathematics, Vol. 9 (2000), 197-219
PoVRay Sources
Minimal Surfaces 