I knew my 2 year old son was being too quiet, when I came into the computer room I saw this:
A quick google search told me that nail polish remover or Mr. Clean Magic Eraser should do the trick. This is the desk after the nail polish remover.
Still not great, but better. I used the Magic Eraser after the nail polish remover, but it wasn't able to help much more. I'm thinking I might be spray painting this desk eventually.