A minimal surface with only catenoidal ends is usually called a k-Noid. The simplest examples are the spherical k-Noids that increase the dihedral symmetry of the catenoid (sacrificing embeddednes) and the toroidal k-Noids with add a handle to the former.
Above are k-Noids of genus k with k-fold dihedral symmetry.
Minimal Surfaces 