My sister cut hers so badly that my mom had to shave her head! They weren't sure she was going to have any hair when she started kindergarten, but thankfully she did! My mom said it actually seemed to grow in thicker, so maybe it was a good thing! She also cut my doll's hair and anything else she got her hands on.
This reminds me of a great scene in the book "Papa's Daughter" (a true story), about a girl (telling the story, as an adult) who loves to cut everything she can get her hands on. She finds a paper and cuts a beautiful deer out of it. She shows it to her parents, proud of her skill. Her father (a minister) wasn't too happy, as it was his sermon that she had cut up!