As an extreme case of the Enneper surfaces with one or two ends that are intrinsically rotationally symmetric there is a translation invariant surface with one periodic Enneper end and one annular flat end.Enneper-Planar

Its associate family continuously translates the surface. It is not embedded and hasn’t caught much attention. Its first appearance might be in a paper by Daniel Freese and Matthias Weber. The Enneper-Weierstraß representation is very simple:

G(z) = z \qquad\text{and}\qquad dh = dz

The surface belongs to the class of translation invariant complete minimal surfaces which have total curvature -4π in the quotient. Other examples are not only the helicoid, its associate family, and the singly periodic Scherk surfaces, but also the half-twisted Scherk surface and its conjugate. I expect these surfaces to be classifiable, but it will require some work.

Resources

Mathematica Notebook

PoVRay Source

D. Freese, M. Weber: On surfaces that are intrinsically surfaces of revolution, Journal of Geometry 108 (2017), 743–762.