oldbobcat

Mileage. This chain is worn out. Note how the chain and the cogs mesh well at around 5 o'clock, and how it rides higher on the teeth as you move away from that position. A worn chain elongates. As the pins and rollers wear, the chain gets longer.

oldbobcat

If you did that you'd wear a hole in the derailleur. That makes it shift even worse, and then you have to buy a new one.

The limit screw is only there to keep you from flipping the chain off while you're shifting. The derailleur height only affects the accuracy of the shift while you're shifting. When it's too high the flexibility of the chain comes into play. We want to minimize that effect. Optimum is 1-2mm. There are malignant cases where you'd want to go higher, but there's no need to go into that here.

urbanknight

That's not exactly what I was suggesting, I think Kontact understood what I was saying, that if flex is causing the chain to jump off, a derailleur that does not leave a large gap between the cage and the chainring would prevent the chain from falling off the outside. I could still be incorrect, but you seem to be implying that I am suggesting the cage should be in constant contact with that chain or something, and I am not.

urbanknight

Looks like I responded while you were adding this. You're saying that both adjustments are only a factor while shifting, but I was wondering if the bottom bracket shell flexing (i.e. during a sprint or climbing out of the saddle) could play a part. Perhaps not.

Kontact 

The chain is worn out. Likely the cassette and possibly chainrings are, too because the chain is so worn out.

But the chain is worn out.

oldbobcat

Come to think of it, many many years ago I had a miserable Raleigh Super Course that would give me chain rub when I stood hard on the cranks. I never knew whether it was the bottom bracket spindle, the crank spider, or the chain stays flexing enough to deflect the chain. It seemed to go away when I put a Stronglight 93 crank on it, but that bike was pretty whippy. I used to also hear of chain stay flex on some steel bikes that would cause the rear derailleur to shift. That would be with friction shifters. But I never dropped a chain that wasn't the result of a botched shift, damaged chain or damaged chain ring.

We did have a lovely CF LeMond with SRAM Force on clearance that would not stay sold because the front derailleur never shifted well. This was before Red, so it was top-drawer. Finally, we figured that somehow the carbon crank spider was bent. We swapped the crank and made someone very happy.

CrimsonEclipse

Yikes!
You need new drive train, chain, chainrings, and likely cassette.
Also check the cogs on your derailleur.

(might be time for a 1x conversion)

Yan 

You're tightening the lower limit screw, not the upper limit screw. You got the two screws mixed up.

Yan 

Your derailleur is too high. Move it down so it's a bit closer to the chainring.

But more importantly, both your chain and chainring are toast. You need to replace both. Cassette too I'd bet.

oldbobcat

And the pulleys on the rear derailleur.

alcjphil

Looking at the photos I am not surprised that the chain comes off the large chainring under load. Where the chain meets the large ring the teeth are not really engaged with the chain. The tiniest amount of flex of the chainring at that point will derail the chain. It is probable that the cassette is also worn out. The photo shows a severe neglect of basic maintenance. The chain should have been replaced a long, long time ago. By this time, cassette and chainrings are also compromised.

ARider2

Exactly what everyone else just said above but probably needs all cables as well given state of deferred maintenance on the drivetrain.

grumpus

Chain and/or chainring worn beyond usable limit, probably cassette too.

veganbikes

Good good but stomping on a bike is bad. Yes having to climb out of the saddle and pushing hard is one thing but stomping is another.

A clean bike is a happy bike but especially in your drivetrain all that grit and grime wear things out faster. Using a hose isn't always bad using a pressure washer isn't ideal (unless a pro with a mechanic) but a normal lower pressure hose and just being mindful will not do much damage. Pouring lube on top of lube on top of lube that isn't great unless maybe using was lube but in that situation still wiping it down is crucial. If your bearings are unsealed then yes a hose probably could wash out the grease but it is rare for a bike these days to have completely unsealed bearings and in some cases they use lesser seals for say a road bike as they don't intend that for much rain usage but a bike like yours probably not using high end bearings like that so no need to worry.

The Crux is a cross bike, cross bikes are designed to get muddy. You can see some cross racers at the end of some races and they are covered in mud like full facial mud with glasses covering the eyes and some of the bikes you see are more mud than bike by the end of it.