According to a variety of authorities, body odor is the result of bacterial interaction with sweat and oil excretions of the body. The medical name for severe body odor is bromhidrosis. But the key is that there is an interaction between the microbiome in the area around your sweat glands, and your sweat that causes the odor.
Anthropologists studying primitive peoples have found many cases of people who do not bathe regularly and yet do not smell either. A group at MIT did some studies with repopulating the microbiome with ammonia consuming bacteria from soil as a way of avoiding body odor without washing. And the folks at the Microbiome project cultivated a bunch of microbiomes from people's arm pits and thighs which also informed this area (although there isn't really a pro-biotic deodorant that I know of yet)
As I understand it, showering and washing isn't getting rid of "odor" it is getting rid of bacteria that is outgassing odor after processing your sweat. You could be odor free without showering with the right bacteria consuming the sweat.