A very simple, fragrance free baby soap, or a goat's milk based soap. I have a friend from HS that reacts to almost every soap, and the one she didn't, isn't made anymore. I had to do some looking, but I'll share what I found:
MONICA'S SOUTH MOUNTAIN SOAPWORKS is a little cottage biz out in Nova Scotia, I found it on FaceBook... she provided me with an ingredients list via e-mail, took the payment via PayPal, and shipped to my front door.
My HS pal now orders directly herself, but was delighted with the product. At $4 a bar or so (Cdn) it's no more expensive than a lot of name-brand soaps.