I did this experiment a couple years ago as well. Tried it for several months. Even tried it with my entire body. Just used water and a wash rag. It went pretty well. The bacteria on your body level off and you eventually stop smelling bad altogether. My wife verified this with her nostrils. What eventually got me back on soap is I began to develop skin problems.

I started getting nasty rashes and eczema, as well as flaky red skin under my beard. Went back to soap and shampoo since but it was an interesting experiment. I only use organic, chemical free products nowadays. I get enough chemicals in the air walking down the street. Don't need to be rubbing it all over myself as well!