I use Whole Foods 365 fragrance free body wash, shampoo and conditioner. For a moisturizer I use Alba very emollient original unscented, for face I use Body Merry, night cream along with Cosmetica pure hyaluronic acid serum and the blue unscented type of chapstick. For a leave in conditioner I use my kids Little Twig unscented conditioner. They also sometimes use the Equate kids 3 in 1 unscented body wash/shampoo/conditioner. For deodorant I use a small bit of Secret unscented, it has a masking fragrance which is noticeable to me but the least offensive unscented deodorant I've found, and I've tried a lot of products. I also like the Alba leave in conditioner and styling cream which are unscented but possibly discontinued. I hope this helps!
Not sure which country you are in but my girlfriend has problems so I've checked the options in the UK a fair bit. There a quite a few products labelled 'sensitive' and unperfumed in the shops here eg. shampoo, washing liquid and so on which are mostly good. Some other products I've failed on eg. bio washing liquid, toilet cleaner.
I have sensitive skin and I stopped using shampoo last winter. I was worried about greasiness but it was no trouble at all.
I use baby shampoo for everything. It's face wash, body wash, shampoo, shaving gel, etc, and has no perfumes.