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