I use eggs nowadays for hair, from pasture-fed chickens, and it gives you some of the "clean" feeling you get with schampoo. It even lathers a bit. But you need to rinse with quite a bit of water, cold water. I have long hair and I wash it once per month. I brush it everyday, that spreads the fat from the skalp evenly in rest of the hair and makes it easy to manage. I always braid my hair, othervise I would not be comfortable with the oily look you get when you don´t wash it oftenly. I never put any other oil in my hair, or on my body, I feel that it makes the skin dry after some time. Just avoiding soap takes care of moisterising skin.
I have never used deodorant or perfumes, can´t stand then, except musk oil wich I find very attractive. Not sure if it is acctually real oil from the musk deer, probably not I´m afraid.
No tothpaste, I brush if I get fuzzy teeth, usually in the evenings. I still use soap (some herbal, coconut stuff) for my armpit, othervise I stink, but I think it might be a problem that will go away after beeing on RAF for some more time. The person I know who have done this the longest (about 5 years), he stank for the first 3 years (detoxing from bad SAD) and is, I would call it, odourless nowadays. No bad breath either. I hope to reach that stage. Don´t mind normal body smell though, kind of like it.
For washing I use the soapnuts that Nicola mentioned earlier, and Raw mentioned Ritha, it´s the same thing (ritha is the name for it here in India and Bhangladesh). It grows where I live, but if I where back in Europe I would prefer to use soft soap from pine trees, or something else local.
In Kashmir I am lucky to have a natural stream just next to our land, with cold glacier water running, and we use it for drinking and bathing. It is rich in bacteria and oh, so tasy. When here in Goa we suffer with the clorine water like everyone else. We take baths in the ocean though, it is better than the tap water, but I dont like salt water hair so I rinse it in tap water.