Sometimes, the Enneper surface will just show up. For instance, when classifying complete minimal surfaces of small total Gauss curvature, it is unavoidable. Together with the catenoid it hold the record of having only total curvature -4𝜋. Next comes -8𝜋, and for this you will encounter critters like these that have look like an Enneper surface with two catenoids poking out.
There are many others, and I view them not so much as objects to be classified and put away but rather as play grounds where one can learn what design goals are compatible with the constraint of being a minimal surface.
For instance, adding a base to the surface above is possible but pulls the two top “lobes” of Enneper and with them the two inward pointing catenoids apart:
But still, the Enneper surface comes in handy. The k-Noids, which traditionally are minimal surfaces just with catenoidal ends, have to be well balanced: The catenoids pull and push in the direction of their axes, and get boring after a while. The Enneper surface is much stronger then any number of catenoids and will win any tug-of-war.