What is the maximum tooth sprocket you could fit on the back using a Deore DX rear derailleur?

setting H and L screws should have no effect on the derailleur beyond what they are designed to do - ie limit how far inboard and outboard it can move. If something has caused damage its far more likely its because one of those limits is set incorrectly and the derailleur has crashed into either the frame or (far more likely) the wheel spokes - think you would notice that however!!

Another thing that could break it is overly tight chain tension (ie incorrect chain length or obviously over capacity setup where it can't take up the slack properly or gets stretched out too much)

I think this is another point where some photos of your setup would MASSIVELY help anyone see what might be up - I think if you put your mind to it you could sort some out :)
 
sounds like the chain wasn't routed properly between the jockey wheels and the cage maybe? (or buggered links that stick out can cause catching issues also)
 
also - and I get you want to do this stuff yourself, but...a visit to the local bike shop might be an idea :)
just have someone suitably qualified take a look
 
Well that was an hour I’ll never get back, hoping there was a happy ending as to the fitment of massive cogs.
Here are my examples;
I recently bought a 26” wheel Cotic Bfe and rode it yesterday for the first time. 1x11 all XT M8000 with a whopping 40t on the back. Can’t remember the front but will check. The first three gears were totally useless, even uphill and tired. Legs spinning like crazy and 1+2 were moving the bike so slow it was hard to balance it and stay on. The mech is obviously sitting a lot closer to the ground which clearly isn’t a good thing. Not in my opinion.
A key one for me is - on another bike I’m building I put a short cage M735? XT r/m to do good old 11-28 and 24/34/46 the other day and when I realised it wouldn’t work I gave in and bought another mech. I could have just accepted that I’ll never use big/big or small/small but you never know and I didn’t like the idea of either accidentally ripping off the mech or having the chain so slack I could run over it.
My Five on 26” wheels runs 2x10 with 11-34 cassette. I use most of it. For me personally that’s enough on that particular bike. The mech is a long cage but doesn’t need to be, I just don’t have a shorter one and not buying one for the sake of it.
My Cotic Soul runs 1x9 using 32/11-34. I only muck around on it so it’s great for my needs. Works perfectly, despite being old.
My Karakoram has a full 27speed M570/M750 group which is great for that bike and its use. Correct sized cogs and mech.
I’ve never tried a mech hanger extender, I didn’t even know they existed.

While I noticed that the bike when I got it had new brake cables it seems the gear cables are old so perhaps that is the issue. The outer casings look good and not worn but looking at the exposed cable it is dirty and rusty so could be decades old. Could that cause indexing issues?
Yes. Even cables that look good are often not. On a mission like this you 100% need brand new quality cables as the very first step.
I notice people like to discourage me from tinkering as they think I am not qualified to do it. They think I not able to process information and will end up doing nothing, which is not accurate. I bought a bike and am improving it. Nothing complicated about that but people say I am 'over analytical' for asking questions to improve my bike.

Since you have added 40t to your old bike I see we are similar! Nearly everyone else discourages such experiments!

Yes I am sure extension would stop it but was trying to avoid using one and use as last resort. If that is the best idea then I will use it but I wonder if there is any problem with how it is currently with the cassette just being very close to the jockey wheel.
I like to push the boundaries but the square will never go in the triangle hole without significant changes such as making the square into a triangle.
I have switched to 2x now anyway since I never used the big cog on 3x anyway. Never understood why you would want to pedal when going downhill.
In most cases, to go faster, in order to get up the other side. To get more air off jumps. To swallow more flies.
I no longer need to hope, I have achieved it and ridden it. I have tasted the fruit of the gods.

I did run the numbers and 22/40 is almost identical to 20/36.

I find the pushback about low gearing peculiar. I don't see people complaining at people riding e-bikes which takes away much more of the physical aspect. Seems some subconscious issues about all the low gear shaming. Maybe e-bikes are so far removed as to not be a threat.
Then you’re not trying hard enough. Search the forum - there’s plenty of dislike towards E-bikes around here.
Oh ffs!!!

I am now in a world more trouble.

I was following the park tool article for setting the H-screw.

I got it to what seemed correct and was going through the indexing process as per the guidance.

I got everything what seemed close and as I went to take it for a ride suddenly the derailleur kept catching on itself as I pedaled.

I then went back to base to take a closer look and it looks like the derailleur has now packed in somehow! I thought the cage might have gotten bent somehow. Maybe too much tension when doing the H-screw exercise? Really did not have time to give it proper inspection in the last spurt of light.

When pedaling the bottom jockey wheel will not take up the chain.

I looked at the hanger and I am not able to tell really if it is that. I don't see how that could bend in that pedestrian process I was just doing. I didn't drop it or 'owt.

It was going dark and had to pack everything away at peak frustration not having got to the bottom of what has happened, having cocked it up farther than ever before and the light dwindling.

Oh FFS indeed.
No disrespect, but this is a total waste of time. Get the right kit for the job and be done with it. When you’ve accumulated 10+ years of parts and a heap of bikes you can take time out to muck around trying new ideas.
You seem to have discovered that when in the higher gears the top jockey wheel is too far from the cogs to give a smooth shift. The derailleur and cassette are designed to work together to avoid this. Let’s say you change your front gears. You have moved the chain over by 6mm or so. That has no effect on the rear. Why? Because it’s so far away. See what I mean? Enough people have asked for photos now to see exactly what it is you’re up to and you really ought to share some.
The most important thing is to enjoy the hobby and I can see you are doing that, which is a good thing, but some things are better left alone.

Edit - spelling mistake.
 
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