In days of old, I amassed a collection of tokens to be used as miniatures for the RPGs I was running (science fiction and fantasy, mostly). I bought sheets of 1-inch-round labels, on which I would print faces, monsters, etc., which I would affix to 1-inch round wooden disks from an art supply store. Some of the wooden disks were basically squared on the edge, which was perfect. Some of them, however, were a lot more rounded and smooth, which made the label crinkle at the edges.
Recently, I acquired a laminator as well as a 1-inch punch. I think what I may do now is experiment. I have full-sheet labels, so my plan is to print a bunch of 1-inch portraits/images, affix the label to a letter-sized piece of cardboard, and then use the punch to liberate them. I may also see what I can do to seal them with acrylic or something like that.
Another option may be to print on some cardstock, laminate it, then punch it out. That would expose the edges to possible damage, but the image would remain protected.