Of course the chemicals will cause damage to her hair. Thirteen is far too young to start down the hair coloring road. The box color you get at the grocery is harsh and will turn her hair into straw in no time at all. Professional coloring at a salon will cost at least $50 and up, and has to be done every four-to-six weeks. Girls who are 13 "desperately" want to do a variety of things; this doesn't mean they should. Instead of accepting your daughter's desperation, it might be wise to explore why it is she thinks her current hair color isn't satisfactory. There's probably more going on than has to do with the color of her hair.