It isn't a culinary application, but egg whites make great facial masks. I read this in Seventeen magazine years ago and still do this today! An egg white mask will temporarily tighten pores and skin for a mini-face lift. Simply whip up the egg whites and spread over cheeks, forehead, nose and chin. Add a dollop of honey for some moisture, or some scented rose water or lavendar for a nice, relaxing scent.
(Editing this rather than adding a new answer, just to keep the non-culinary applications in one place.)
Another non-culinary application is glair for painting - a variant of watercolor. Whip an egg white until very stiff (way past the "stiff but not dry" advice of cookbooks), then let it sit until it re-liquefies. Remove the crusty stuff from the top, and put the rest in a clean jar. To paint with it, combine the glair with your choice of powdered pigment in a small palette (I use clam shells), using just enough glair to make it flow from your brush. It can help to add just a touch of honey to keep the paint film more flexible. If needed, thin with a little water. Note that the older the glair, the better it works. It will get stinky, but not rotten-egg stinky, because there's no sulfur in it.