One can add vertical handles to the Karcher-Scherk surface with at least 6 ends. This possibility was first mentioned in Hermann Karcher’s Tokyo notes.
Above are two 5-ended versions that have the symmetry of a prism over an equilateral triangle. They come in a 1-parameter family with varying angle between the ends. There is no doubt that less symmetric versions can be made.
I also suspect that some of these surfaces can be twisted like the helicoidal Karcher-Scherk surfaces so that the annular ends become helicoidal and the surfaces stay embedded. It would be interesting to know the limit of such a deformation.