Inspired by this thread:
What is the longest time an engine has run continuously, without turning it off for routine maintainence or repair or refueling?
I’d like to restrict this answer to internal combustion or steam engines, since electrical motors are basically maintainence-free items which can reliably run continuously for years.
This is unlikely to be the absolute record, but is probably worth mentioning:
From this link.
How long does a nuclear power plant go between shutdowns? It’s just a massive steam engine attached to a generator. What about an aircraft carrier on deployment?
I’m a lot more curious about how long an internal combustion engine has run without stopping. Xema’s post details a quite incredible feat given the length of time most vehicle engines run.
I suspect if it is limited to internal combustion then the winner will be some form of a diesel or fuel-oil engine. Although for the life of me I can’t find anything on Google that tells me what kind. My WAG is a cargo or bulk-transport ship or a locomotive, but based on size and necessite I’m going to go with the ship.
Many diesel engines in the arctic never get shut off or so I have been told. I have no idea how they change the oil. 
Not a winner, but worth a comment:
In the early 1950s to combat rumors that their twin cam engine was “weak” and prone to breakage, Jaguar put a couple of XK120s on the track for one week straight. They stopped for fuel, but I don’t believe they were ever shut off. At the end of one week, the engines were torn down and examined for wear. No wear was found.
After watching Ice Road Truckers I considered this might be possible, but then I remembered that they said the trucks have to have their oil changed after about 10k miles or 1 month.
But then again, big rigs are not the only diesel engines up in those parts, and now that I think of it I seem to recall that the show said the power plants at all the diamond mines up there are all diesel powered, so perhaps I should change my somewhat-educated WAG to that. Those mines need insane amounts of electricity to keep running and I’m sure they need those generators running as much as humanly possible to satisfy the load.
If steam engines count, does that not include steam turbines? A turbine would be the simplest of the engines, basically with nothing to wear out that I know of, except a bearing. My guess would be it would be a steam turbine, probably one at a power plant.
When I worked on the oil patch I ran a wireline truck that I didn’t shut off for a month and a half
I used to work at a natural gas compressor station, some of those engines would run for over a year between maintenance shutdowns. There were 4 straight 7’s and one V-8, the pistons were the size of beer kegs, and they’d run at around 300 RPM.
Big stationary piston engines can have oil circulating outside of the engine to tanks or other means that could be changed, or continuously replenished and traded out, while running. I’d bet they do.
Large engines such as generators have lu oil tanks which have oil circulated by oil pumps.
These tanks can be filled anytime.
I would expect the answer is probably a pump or generator of some sort.
Hesitant as I am to bring back a zombie thread, especially on my first post, I would like to suggest a Bessemer hot tube engine that has been running in a Pennsylvania oil field, without shutdown, since 1899.
OK, that’s a good first post, but do you have more information - like a link?
(Typically people here will just say “cite?”)
Apocrypha.
I’ve heard the story. I can even link to somebody else mentioning it online.
Frankly, I found this thread trying to find if there’ve been any that ran longer.
Assuming it’s true, I’m hopeful somebody closer tracks the engine down and writes an article on it.
Would a natural gas, or landfill gas, flare off be a very primitive inefficient engine, in that it would provide some thrust (however unusable in they way we use it).
I’d consider one of those to be a fire, not an engine.
NASA ran an ion propulsion engine for over five and a half years, setting a world record for the longest test of a space propulsion system last year.
Some of the largest engines on the planet can apparently be serviced while running.
Those building-sized Wärtsilä engines that power cargo ships may be capable of near-indefinite running as they’re loping along at a stately 90-100 RPM or so. Bearing wear has got to be somewhere beyond negligible at that “speed.” For comparison, most car engines idle at around 750-1000 RPM.
And I believe some of those large engines have the ability to shut down a single cylinder for repair
Something that isn’t an engine but might be interesting:
There is some debate and no shortage of [del]utter nonsense[/del] maritime lore along the lines of “We were running dead slow, so the third mate could crawl in with a wrench and give the nut a turn each time it came by” on that.
It’s possible to “lift” the valves so the cylinder is dead, but slow or not, the pistons, connecting rods, etc. are still in motion. This makes changing bearings or piston rings impossible, but it is possible to work on the top end to do things like replace valve seals or fuel injectors. If you’re facing a more dire situation like a broken crosshead, or piston, it is possible to stop the engine, disconnect the broken bits from the crankshaft and secure the piston, which will allow you to restart and proceed minus that one cylinder.