In the same 1982 paper in which Chi Cheng Chen and Fritz Gackstatter proved the existence of a torus with a single Enneper end, they also construct a genus 2 example and give the Enneper-Weierstraß data for a genus 3 surface.
This is the beginning of an infinite sequence of finite total curvature surfaces, indexed by the genus, with one Enneper end of winding number 3. Edward Thayer has computed numerical examples of these surfaces for high genus as well as for their dihedalizations.
The Chen-Gackstatter surface of genus g has finite total curvature -4π(g+1), the largest possible value for its genus. It is conjectured that these surfaces are the unique in this respect. It is also conjectured that for increasing genus, they converge to the singly periodic Scherk surface.
The existence of the Chen-Gackstatter surfaces was first established by Katsunori Sato in 1996 and independently by Matthias Weber and Michael Wolf in 1998. The latter proof is based on the observation that the existence of such surfaces depends on the existence of polygonal zigzags that divide the plane into polygons that are conformally equivalent under a vertex preserving map.
C.C. Chen and F. Gackstatter: Elliptische und Hyperelliptische Function und vollständige Minimalflächen von Enneperschan Typ, Math. Ann. 259 (1982), 359–369.
K. Sato: Existence proof of One-ended Minimal Surfaces with Finite Total Curvature, Tohoku Math. J. (2) 48 (1996), 229–246.
E. Thayer: Higher-genus Chen–Gackstatter surfaces and the Weierstrass representation for surfaces of infinite genus, Experiment. Math. 4 (1995), 19–39