The simplest minimal surface (besides the plane, of course) that contains a circle is the catenoid. Here, the surface normal makes constant angle with the plane of the circle. Rotating the normal with constant speed along the circle gives an explicit family of circular helicoids.
With faster spinning normals, the surfaces converge in solid torus about the circle to a foliation of that torus by disks perpendicular to the circle, with the circle being a singular set of convergence.
The image is deceiving: All these surfaces have two ends and finite total curvature, with limiting normal perpendicular to the plane of the circle, just like the catenoid. Of course, they are (except for the catenoid) not embedded, but outside thin cylinders they are.
A particular case arises when the normal spins an odd multiple of 180º around the circle. For a single spin, one obtains Meeks’ Möbius strip (which was discovered differently) above. Next to it is the 5-fold twisted Möbius strip.
P. Mira: Complete minimal Möbius strips in Rn and the Björling problem, J. Geom. Phys. 56 (2006), 1506–1515.
W. Meeks, M. Weber: Bending the Helicoid