Chen and Gackstatter constructed two complete minimal surfaces
of finite total curvature, each having one Enneper-type end and all
the symmetries of Enneper's surface. Karcher generalized the
genus-one surface by increasing the winding order of the end. We
prove that a similar generalization of the Chen-Gackstatter genus-two surface also
exists. We describe a collection of immersed minimal surfaces that
generalize both Chen-Gackstatter's and Karcher's surfaces by increasing the genus
and the winding order of the end. The period problem associated with
each of these surfaces is explained geometrically, and we present
numerical evidence of its solvability for surfaces of genus as high as
35. We also make conjectures concerning these surfaces, and explain
their motivation. Our numerical results led us to the Weierstrass
data for several infinite-genus, one-ended, periodic minimal surfaces.