Adding a handle to half-fence of catenoids requires to solve a 2-dimensional period problem. The resulting surface is the limit of k-Noids of genus k for k to ∞.
If we allow the annular ends to become non-parallel, we obtain a 1-parameter family.
The same works for two (and likely many more handles). Below is the genus two surface with parallel annular ends. This one is likely the limit of k-Noids of genus 2k for k to ∞.
Resources
Mathematica notebooks for genus one and genus two (by Ramazan Yol)
Minimal Surfaces 