If you're serious about
cheapest as the most important variable, then the sort of glib (but true) answer to this question is to observe where people who have
no other choice end up living, which means a trailer in a rural area or housing in the cheapest parts of any large U.S. metro.
If you're lucky enough to have startup capital, then you can reduce your recurring expenses to practically nothing. You can find a trailer for a few grand, for example, and functional houses can be found for less than $20k in many American cities. You can probably find parks charging only a couple hundred a month for lot rent, if you go the trailer route, and property taxes will be minimal on houses in the described range, if you choose that, instead.
If you don't want to invest anything up front, you can still find housing for around $350/mo in the worst parts of the Midwestern city I live in. (But you shouldn't expect a very hands-on landlord.) You might even find weekly- or daily-rate housing that's cheaper, but I've never personally looked into that sort of thing.
Obviously, there's a reason that these areas are so inexpensive, but I've optimized only for cost.