For me, I think the record is probably about 250 miles, from somewhere in Indiana where I stopped for supper and Ashtabula, Ohio, where I spent the night. I know that it was over four hours because my Suburu Forrester will announce every hour how many hours you’ve been driving nonstop. That was an 800+ mile day, which seemed like a good idea at the time.
It was like 1600 miles or something. About 26 hours. Rained hard about half of that. It was pretty truly miserable. I was hauling livestock and needed to keep them wheels rolling to get them to where they were going. Only redeeming thing was I just installed a stereo with a USB port, so no having to screw around with constantly changing CDs or radio surfing. Just plug and drive… and drive… and drive…
Not counting gas or pee breaks basically border to border. 1275 miles from north of Burlington, WA to where I live in SoCal. I used to do that run twice every summer around the 4th of July.
A couple of years ago I drove home from Cape Canaveral, to DC in one night, roughly 900 miles. I don’t remember exactly how long it took, but I left in the late afternoon and pulled into my driveway at about 0700.
Another time I drove from DC to Atlanta, about 650 miles. Supposed to take about 10 hours; I did it in 8. Almost made it all the way on one tank in my Crown Victoria. Had to get gas about 10 miles before getting on I-285 (the Atlanta beltway). I distinctly remember zooming past a speed trap in North Carolina, and waving at the cops. They just waved back. 
New Orleans to Los Angeles straight thru (I-10), sharing the driving with my spouse, with stops for fueling, bathrooms. About 28 hours.
If the OP means literally not stopping for any reason, even gas and bathroom breaks, I have no idea as I’ve never kept track of that.
If we mean only minimal stops for fuel and bathrooms, when I moved from North Carolina to California, I drove the whole way. On the first day I drove the Saturn I had at the time, and my dad drove the rental truck. We started out in the morning a bit north of Charlotte and ended the day in Little Rock, however many miles that is. It rained the whole day, so neither of us wanted to get out our vehicles unless we had to. I’m sure we did stop at some fast food places for lunch and dinner, though, plus breaks for bathrooms and gas. That day, I’m pretty sure, was the longest day of just driving.
On the second I started driving the truck after lunch and from that point on we started swapping vehicles after each break. I found that made a huge difference in terms of comfort, with the different seating position in the truck versus my car. Much better than just sitting in the same position all day.
With no stops at all, I’ve done the trip from my home in suburban Chicago, to my parents’ home in Green Bay (about 200 miles) nonstop a number of times; however, as I’ve gotten older, there’s more often now a restroom stop in there somewhere. 
With stops, the longest one-day drive I’ve done has been around 850 miles (Oklahoma City to Flagstaff) – granted, my wife and I traded off stints behind the wheel, and we stopped several times, but that was a loooooong day in the car.
Per the OP’s description of “non-stop”, are you saying you never parked and got out of your vehicle until you had covered 1600 miles?
Are we really being sticklers for this, because if we are, the maximum answer for just about everyone is going to be about 300 miles. Somehow I doubt that’s what the OP meant.
I think that’s exactly what the OP meant.
The longest non-stop driving I ever did was between the towns of Saltillo and Zacatecas (232 miles) in Mexico. It was memorable because there were no gas stations between these two towns and I worried about running out before reaching the next (the Green Angels patrol this long stretch pretty regularly for this very reason).
The longest day of driving with bathroom and gas stops was between Ft. Collins (top of Colorado) and Trinidad (bottom of Colorado). The trip started after it started snowing while eating breakfast at an IHOP. I jumped in the car to avoid a coming blizzard. By the time I hit Denver (a third of the distance), it had become a full blizzard, but I was determined to make it across the Raton Pass into New Mexico before stopping. The remainder of the trip was me driving a Pontiac Sunfire at 25-30 miles per hour on a snow-covered through the ruts left by semi trucks. I was forced to stop due to the Raton Pass being blocked off due to the snow and was able to get the second to last hotel room in town where I spent a restless night.
We do 200 miles w/o a stop quite often, from home to a vacation place.
Including gas stops, back in college, my wife and I drove about 1200 miles straight thru from Champaign, IL to Ft Meyers FL. Like an idiot, we headed out on a Fri afternoon, after a having a few beers. As I recall, Atlanta construction, in the rain, in the middle of the night, was NO FUN!
Earlier this year, we did 1000 miles from Chicago to/from my kid in Denver, w/ only gas stops. No big deal. We’ve done similar long shots to/from CO and SD maybe 5 times. Our preference is to start ridiculously early - say 4-5 a.m., and you get in by 7-8 (absent time zones). W/ 2 drivers, plenty of food and listening material, we’d generally prefer one LONG travel day to 2 shorter ones. And we don’t consider a long day of driving all that much more demanding than air travel - with driving and time on both ends.
Well, lessee, the longest I can remember was about 420 miles from Paisley OR back to Portland–the gas station in Paisley sold non-ethanol gas and that gave the Camry I was driving at the time quite a bit of a mileage boost over the usual 325 miles per tank. I took the scenic route on that one, usual route is a little over 300 miles but I wanted to see some stuff and took a lot of back roads. I once did Sacramento to Los Angeles (about 375 miles) on one tank in an itty bitty Chevy Sprint back in the day.
If we’re talking with gas breaks, I did Portland to Green River UT in one stretch and that was about 950 miles. I gassed up in Ontario and Salt Lake City I think. Then 70 miles in a kayak that took three days. Then Moab to Yellowstone Park (about 550) miles with a leisurely break for the night in the middle. Then all around the park and back to Portland in one stretch, that’s about 850 miles. That was one helluva trip.
Roseburg, Oregon to Bigfork, Montana ca. 1998. I was 17. Took me about 15 hours IIRC and Google is telling me it was 766 miles. I stopped for gas half a dozen times and once in Kennewick, WA for McNuggets. Otherwise it was a straight shot… in a 1975 VW hippie bus. I had a box of mullet rock cassette tapes and a 6-pack of Pepsi to get me through the day. Good times.
689 miles, Hughson, CA to Eagle Creek, OR, but technically was not non-stop - there was a 10 minute fueling stop in there somewhere. Made the trip several times - leave home in Eagle Creek headed south, stop & sleep 3 or 4 hours at Klamath State Park, head to Hughson, fuel up there, load my game birds, and be back on the road. Best time for me home-to-home on that trip was 27 hours round trip.
I drive from Orlando to Miami and Tallahassee (about 240 miles and 260 miles, respectively) non-stop regularly for work. I once drove from Orlando to Pompano Beach and back to pick up my college girlfriend without stopping, except to let her into the car. That was 400 miles almost on the nose.
If we are not counting gas/pee stops as “stops,” I’ve driven ~900 miles from Orlando to Washington DC (or vice versa) non-stop a few times. The worst trip was helping my friend move back to Orlando; she rented a Ryder truck with no cruise control, and I had to drive for about 15 hours because it wouldn’t go much faster than 60 with a full load of furniture. My leg was cramped in a bent position for about 24 hours.
My personal record (no stops of any kind) is Fort Worth, Texas to Deming, N.M. 700+ miles. I just managed with empty Gator-Ade bottles, lots of water and being a guy.
I’ve posted before about my various vehicles with auxiliary tanks. My old truck for the above trip had enough tanks to go about 850 miles. My current truck can easily top 2000 miles non-stop, but 60-yr-old me is less enthusiastic about pushing those limits.
As far as I can recall, true no-stopping straight shot must have been about 2.5+ hours over a bit over 150 miles, Barstow-Vegas. Longest with only one fuel/toilet stop with no eating or “rest period”, about 5 hours and 280ish miles (BWI Airport in MD to The Greenbrier in WV). Longest single-day: DC to Rutland, VT, 8-9 hrs 500 miles.
With stops I did the entirety of I-40 + a bit in 3 days. San Diego to Wilmington, NC. I think I stopped somewhere in New Mexico and someplace in Arkansas.
Longest one day of driving was Atlanta to Ithaca, NY, which the Google says is 900 miles. The last bit of I-81 through Pennsylvania really sucked. Probably 1 AM, just wanted to be done. Not that I-81 through PA doesn’t always suck.
Kansas City to around Gainesville Florida. I’m not sure exactly but it took us about 16-17 hours. Dumb mistake we made was stopping for lunch at a sit down restaurant which cost us over an hour because the service was soooo slow. Now, never again. We only stop at fast food and I prefer combination gas station/food/rest room and our goal is in and out with everything in 30 minutes. Also I prefer McDonalds because from experience they have the cleanest bathrooms and the fastest and most consistent service. We prefer to eat inside because its a bit of a break and keeps the car cleaner.