How can someone not take a shower for 3 years and stay healthy?

But why would she get sick from not bathing? You don't have more microbes penetrate your skin to make you sick, and if your skin is healthy, not bathing will not damage it, we know from the tropics where people shower 2 to 3 times a day if done using soap can dry the skin out and cause serious itching.  Just the stink from not bathing will be (very) bothersome, in the middle ages dealt with by using lots of perfumes. In the hot tropics everybody is used to bathing, in Europe however in the middle ages because the church frowned upon it, and doctors though it would cause diseases by introducing these with the waters through the by bathing widened pores of the skin, people seldom took a bath. Some say they only bathed twice: one when they were born, and the second time the got married (the third time after they died, but then not alive anymore). See:  http://www.todayifoundout.com/index.php/2011/05/why-bathing-was-uncommon-in-medieval-europe/ I do agree with that her phobia should be treated by a psychologist or psychiatrist.

I don't think there is any science that suggests, that not taking showers on a regular basis should in any way be unhealthy. On the contrary I think it might be problematic if you shower too often and use a lot of soap, this is science-wise more likely to cause damage than not taking a shower at all. I am not a scientist, but I think the above is fairly correct.

During the teenage years when the A-Typical child refuses to shower and bath they are building up an immunity to illness and disease. Parents assume that they are just lazy but this behaviour is in fact involuntarily programmed into the child's mind and this has evolved over the last 60 years. Scientists have proven that the final level of medical immunity is completed between the age of 13-17. So just let them stay in their bed all day playing computer games, it's good for them!

Water phobia to that extent should be addressed by a psychologist.   Being dirty is not necessarily unhealthy. The body adapts ... however there is a risk of fungal infections when there is constant moist, dark, warm and airless areas of the body. If any of those 4 requirements for fungus are not provided, then it will not develop.   If the clothes are always clean then body odour can be minimised.   There are cheap creams or oils that can be rubbed over the body, then wiped off to maintain hygiene. When I looked after frail elderly who could not get into the bath, or shower, I would "wash" them with "aqueous cream" and wipe it off.  That is the way to not get into water and stay clean.   If the person has a water phobia to the extent they do not brush their teeth or use other methods of oral hygiene then disease is more likely to result.   Does that person smell?

We evolved getting wet when we got wet.  (We wore animal skins to stay dry.)  We didn't have anything like soap for tens of thousands of years after we became "modern man" (or for the millions of years before that). The fact that you asked the question is proof that people can go their entire lives without a shower, or even washing their bodies, for thousands of lifetimes.  (Caves probably smelled like old gym socks, but cavemen would probably be sickened by the smell of car exhausts.  You get used to what you live in.) The only problem is that others might find the odor offensive, but if she washes any cut or any other break in the skin and applies an antibiotic cream, not showering isn't going to affect her health.

Ok. Showering isn't really for the sake of health: it's something we do in order to not smell bad, and not grossing other people out.