2-3 days. A couple of bouts with bronchitis, a couple of weekend camping trips.
Almost three weeks, after I broke my leg at the ankle. Three days for the swelling to subside, operated on on day 4, spend 5 and 6 recovering (surgeon on day 5: “You could go home today, but why not wait till tomorrow?” Home on day 6, then for two weeks visiting nurse to change the dressing. Finally, the staples came out (ouch) and the incision had healed and I can take a shower. Of course, I washed as well as I could in the meantime, but it felt GOOD to take that shower. Fortunately no cast and no crutches after the op which put in a plate and I was advised to put weight on it right away so that I wouldn’t need rehab (I didn’t).
When I had an emergency appendectomy I was bed-ridden for 2 weeks. No food or water (I had an IV), no toilet, shower or brushing teeth. This was back when they had to cut open your abdomen to get to the appendix, and clean everything out, since it had ruptured.
They did, however, allow me to smoke.
Two weeks at least pre-puberty (as a kid I only bathed once per week usually, Saturday nights - and that was because my mom made me), probably a week and change as an adolescent. Recently, every day to every 2 days by preference.
My dad went for over 2 years (maybe 3 years?)without a real bath or shower (he was in a coma, then paralyzed, then by the time he regained the ability to breathe and move a little on his own he had grown all kinds of extra bone in and around his hip sockets so he couldn’t even be bent in the middle to be sat up in a wheelchair and rolled into an accessible bathing facility - plus he had all kinds of tubes and healing surgical incisions for a very long time, none of which should be exposed to running water). After he was out of the ICU he was kept clean in other ways and smelled fine.
Three weeks (backpacking). However, there were some icy dunks. Very brief ones.
Coming back to civilization we stopped at a diner. The regulars could tell instantly. They were used to stinky people who smelled like campfire (at best!).
About ten days on military exercise. By the end, we all lined up for the bus and it rained on us. The ride back to the base was pretty intense smelling.
Shower felt awesome.
Like Kyla I have experienced the Indian bucket+cup method for long periods of time. The longest was probably around 5 months. And while a shower is a wonderful thing after such a run, I can’t help but be amazed that the Indian method basically allows you to get completely clean using less than 5 litres of water. (I wouldn’t do it at home, though, but showering/bathing is such a waste of water in comparison).
Months.
While working.
Sorry military.
In the Army. Went to California and couldn’t shower for weeks. Went to Iraq and had a nice hot shower every day. Go figure.
About 75 days without a running water shower or bath deployed to Saudi Arabia and Kuwait during Desert Shield/Storm. We had decontamination trucks come by our location for two days once, and I was waiting until the rest of my unit had gone through before I did (was Battery XO at the time). They canceled the second day. Sponge baths from a bucket of cold water, baby wipes, and a lot of getting used to it. My first shower after that was truly a thing of beauty.
Something like 2+ months with no washing at all. From mid September until sometime in late November 1969 I was in the Que Son Mountains in South Vietnam. There were no rivers or ponds anywhere. It did, however rain some 300 inches in October.
We entered the mountains in late August but in mid September I went on R&R in Hawaii where the wife met me. Ms Hook has remarked several times since then on how much time I spent in the shower.
heh, while out camping [and not under fire …] I remember one seriously rainy Pennsic war stepping out to go to the bath house for a shower, getting soaked to the skin before managing to walk out of camp, going back to my tent and stripping down, soaping up and walking back outside for a rinse off 
Three weeks in Norway while our battalion was on a NATO deployment. After about week #1 living with 9 other men in a tent things began to smell a little bit funky. After week #2 things were becoming down right nasty.
No need to describe the 3rd week.