If you are not limited to bounded shapes, you can think of shapes that have some translation symmetry; this immediately makes their symmtry group infinite. There are plenty examples of such shapes: a line, a horizontal infinite strip of finite width, a discrete infinite lattice of points.
Basically it is best to think first of which infinite symmetry group you want to have, and then adapt your shape to that. One general method is to take one chose subset of the plane (a single point will do) and add all the transforms of it by the group. For the rotation group this results in such stuff as one or more concentric circles or regions bounded by such circles. These shapes in fact get additional reflection symmetry for free, but that doesn't hurt.
You can have fun with other infinite groups. Think of the group generated by a single rotation by an angle that is irrational to the full rotation by $2\pi$.