Not surprisingly, one can add more than one handle to the Fence of Catenoids.
In the examples above, two handles are placed on the two vertical symmetry lines of the catenoid. The quotient surfaces of this 1-parameter family have genus 3, and the period problem becomes 2-dimensional. I am sure there are more ways to place the handles. Note that the relative size of the handles depends on their position.
Minimal Surfaces 