I had a drive shaft break off.
Also my hood flew up , I was able to pull over without wrecking.
Car Battery was stolen.
Packrats chewed through my speedometer wires.
I had a battery blow up one time when I started the car. I let the water in the battery get low and hydrogen gas had built up in the cells.
Got a hole in my radiator. Which sprayed onto the alternator. Which shorted out the voltage regulator. Which briefly made all the lights blaze like high beams, including the parking lights, brake lights, tail lights, turn signals…
Just after the car is turned off, it emits a weird midrange frequency sound for a while. I have no idea what’s up with it, but it isn’t hurting anything, AFAIK, so I’m ignoring it.
ETA:
When I was growing up, my dad turned over the project of wiring the trailer lights to the car lights. My brother did it such a way that with a certain combination of turn lights and break lights, it would blow the fuse. That make for really tense vacations. Eventually, I took it over and did it more systematically, and we were good to go.
Right after buying my current car, I put a bunch of stuff in the trunk from my old car and started driving around. For about a week I was hearing this strange knocking sound that seemed to be coming from the roof somehow. I couldn’t figure it out until I realized the milk crate of stuff I had in the trunk included one of those novelty mini-Louisville Sluggers that you get at the end of the Louisville Slugger factory tour sitting upright in the crate, and when I was braking, the little bat was knocking against the top of the trunk, which resonated through the roof.
Good thing I figured that out before I started tearing the ceiling upholstery apart.
The radio in my 2012 Jeep Grand Cherokee has always had a weird issue where once every few months, it would stop working. It would turn on, but no sound and the volume control didn’t work at all. If I turned the Jeep off and right back on it would still have the problem. But if I turned off the Jeep and waited a bit, it would work and never stop working again for months.
I had the radio replaced while it was still under warranty, but the problem keeps happening. So it’s some kind of dashboard wiring issue, I assume? It’s so infrequent and random I just live with it, but on the rare times it happens it’s very annoying, because I need to listen to music or podcasts when I’m driving.
I didn’t know it, but I dented my gas tank pretty significantly. As a result, my fuel gauge indicated 1/4 tank when empty. I didn’t often go below a quarter tank, and didn’t notice a problem, until one day my car just “died.” It appeared I had 1/4 tank of fuel, so I had no idea what the problem was. I got a tow to a service station and the car started right up. (gas jostled around during the tow). I drove away, confused, and the the car died within a mile. Eventually, the problem was discovered are remedied.
We had a Ford Escape hybrid that developed the disconcerting habit of suddenly lurching forward just as you were braking to a complete stop (not fun when you were pulling up behind another vehicle at a stop sign or light).
When we took it in to the Ford dealership under warranty, we were first told that it was “normal hybrid operation”. After much back-and-forth they “discovered” that it was a known problem fixable by installing a replacement part.
Knew a guy whose Renault Alliance broke down so many times he gave it away to strangers for free. He gave it to the people who lived at the site of the last breakdown. The cops came to his house to make sure he really gave it away and he said yes he did .
Broke the right front spindle on a 1961 Ford Thunderbird just as I started down a mile long hill on a freeway. When the tire/wheel combination came off, it also took the brake drum with it. I hit the brakes and the pedal went to the floor. While trying to slow down from about 55 mph I was also watching the wheel/tire roll down the hill in the middle of the fast lane. I finally got the car stopped using the parking brake. At the bottom of the hill the freeway made a right turn, on the other side of the freeway was the Western Washington fairgrounds. I found pay phone, called home, then went looking for the tire. It smashed through some bushes and hit a gate in a chain link fence, destroying the gate. A few months later I attended the fair and went looking for the smashed gate. Found a new one with some new fencing. I have always wondered if they figured out what happened.
I had a 97 Accord with an electrical issue where the entire car just turned itself off as I was cruising along going 65 in the middle lane of an expressway.
Radio, dashboard, engine just shut down and I was coasting along in neutral with no power steering.
My ex called me one day because her car was making some type of noise. IIRC, it was late, I head a headache, we were in the middle of a divorce, she was like an hour away from where I was…and I just didn’t care. The noise was loud enough that I could hear it over the phone. I told her to take it to a dealership and let them figure it out.
Turns out there’s a plastic cover over the u-joint, which must have been hit with a rock or something and got knocked out of place. The noise was the cover hitting the u-joint like a playing card in a bicycle spoke.
I had an older pick up I was hoping to put a plow on, but for the year or so I had it, it was giving me nothing but problems. At one point, the thing lived on a charger for months as I was trying to work out why it wouldn’t start. After pulling my hair out trying everything I could think of, in frustration, I drained a snapple bottle full of gas from the tank with the intention of (safely) setting some of it fire. I was so lost for ideas I was jokingly making sure the gas would actually burn*. So, I tap into the fuel rail, start the fuel pump and filled up my snapple bottle with gas. A few minutes later when I walked back over to where I had set the bottle down, I saw that it had separated. It was probably half water.
I emptied the tank, got all the water out, replaced the gasket on the fuel pump, added new gas and it ran fine (for a while, then I sold it because it was getting to be an expensive hobby).
I had a 94 Ford Ranger that would just randomly refuse to start. Crank all day long, spark to the plugs, but no fire. Had it towed to several mechanics when it stranded us, but it would ALWAYS START FINE when they tried to reproduce the problem.
Fortunately, we never had anyone try randomly replacing parts on a problem they couldn’t reproduce. Finally I had a mechanic who by chance stored it outside on a sunny day and presto, he could reproduce it. The fuel pump had a fault in it, and the circuit would go open if you let the truck sit outside in the sun on an appropriately hot day. Get the little black truck out of the sun to cool off or start it early before it got hot, and the fuel pump worked again. After replacing the fuel pump, it never did it again.
I maintain eight, old vehicles. I could write a book on all the car problems I’ve had, though I’m not sure how many would be considered “unusual.”
I suppose one of the more unusual problems was when I was driving my 1997 GMC Safari van. I lost all oil pressure. I stopped and opened the hood. There was oil everywhere. 
I hadn’t noticed it in time and it destroyed the engine.
The culprit? A flexible oil hose had ruptured.
Now some of you might be asking: “An oil hose? Why would it have an oil hose?” Well, the guy I bought it from had ordered the van with a tow package. Included in the tow package was an oil cooler. The oil cooler connected to the engine via flexible hoses. That’s when I realized that oil hoses are ticking time bombs, and to avoid them if possible. (Since I didn’t tow anything, I should have removed it, along with the hoses.)
I know that hill; on SR 512 heading down into Puyallup. That’s a bad place to lose your brakes. Glad you made it through that okay.
My best story was when I was doing some work on my MG. That car has an electric fuel pump, so as soon as you turn the key on, before you even start the engine, there’s a ticking sound coming from the back of the car. I was doing some work on it (can’t remember what) and I heard that ticking sound. I thought it was the fuel pump, but then I figured it couldn’t be because the key wasn’t even in the ignition. But English cars of that era were known for some electrical weirdness. Then I realized that the battery (well, one of them) was disconnected and sitting on the floor of the garage, so it really couldn’t be the fuel pump. I checked the car; one of our cats had climbed in and was purring.
Not quite as bad, but I got a cheap paint job on the old Buick I had in college. As I was driving down the freeway big sections of paint started to peel off the hood, fly off, and hit the windshield. (The shop that painted it did repaint the hood under warranty).
Best I got is one time we had a weird rattle in the dash of the Camry wagon. After extensive efforts to identify, the culprit was an M&M one of our kids had stuck into an air vent. My wife apparently neglected to ask the tech what color.
I once had the gear shift linkage break. It’s a strange feeling when the shifter just flops around not doing anything. Fortunately, it was in second gear, so I could limp it out of the intersection I was in. At the entrance to the mall.
I had catastrophic tread failure on I-94 when I was a senior in college. I’m cruising along in the middle lane at 65 mph when suddenly the right passenger tire just explodes (or so it felt) and the steel belt is just bangbangbanging the wheel well into oblivion. I quickly decide not to hit the brakes and bring the car’s weight forward. I just take my foot off the throttle, signal to turn right, try to guide my car as gently as possible, and coast into the next available exit, where (thankfully), there is gas station right there. I pull into it, easy on the brakes, and exit the car to assess the damage.
My tire is in absolute shreds. My wheel well is half-crumpled. Thankfully, this is in the days every car came with a donut spare, so I jack up the car and replace the tire. It is small enough to clear the jagged edges of the mangled wheel well, and I am able to drive home. Or, rather, to the bar where I was meeting my friends with now an interesting story for why I’m late. A full-sized spare would not have fit due to the damage. And I’m pretty sure Fix-A-Flat would not have done much good. 
(I know Firestone right around that time had issues with catastrophic tread failure on some of their tires, but I do not know whether it was Firestones I had on.)