A level 20 moon druid is by no means unkillable. However, they do have a sack of infinitely regenerating hitpoints they can use unless you either a) break through them or b) have some clever way to circumvent them. The problem is, barbarians have neither. Even at level 19, people are right in pointing out that druid is a full caster class with access to, among other things, call lightning and polymorph (which he can also cast on the barbarian).
There are a number of classes who could potentially win against a moon druid. The fight with moon druid vs wizard has already been done, and opinions are mixed, but it seems majority consensus is that a wizard (especially certain subclasses like chronurgy and divination), could beat moon druid for the majority by forcing bad saving throws.
Druid 20 vs. Wizard 20 - Druid - Class Forums - D&D Beyond Forums - D&D Beyond Druid 20 vs. Wizard 20 - Druid - Class Forums - D&D Beyond Forums - D&D Beyond. Sorcerer or bard could arguably do equally well, especially given the sorcerer's meta-magic.
The problem is that martial classes, in general, are just WEAK late game. Why? Because spellcasting is so powerful and in many cases game-breakingly so. A level 20 wizard can cast wish or true polymorph or teleport or reverse gravity. Or any form of F you. A barbarian can . . . hit things. And soak a lot of damage. That's as it should be. A barbarian or fighter SHOULDN'T be at the level of any level 20 full caster from a realism standpoint. From a gameplay standpoint, of course, you want a certain degree of parity and balance - which you get.
In terms of pure martial (no magic) classes, an open-hand monk is probably your best bet to kill a moon druid, for two reasons. As someone already mentioned, open-hand monks have this ability called quivering palm, which let's them drop a creature to 0 hp. Druids do not normally have super high con or proficiency in con saves. Even their beast forms don't have THAT high of a con, and the druid is looking at a 19 con save to just NOT DIE. That's some pretty impressive stuff. I would have to look more into whether that would just knock the beast form down to zero and knock them out of wildshape, or if it would straight take the druid to zero hp, but either way, that's bad. And if it's the latter, that druid is pretty toast.
Monks also get this nice little thing called stunning strike, which for one ki point, keeps the druid from doing ANYTHING, including wildshape, until the end of the monk's next turn (assuming a failed con save). Of course, this all assumes the druid doesn't take a bird form or cast any particularly nasty spells, but I see this matchup looking pretty good for the monk. At least much more plausible than for the barbarian.
If we assume the great weapon fighting feat someone else mentioned, it might be possible for a fighter to do enough damage to knock a druid out of wildshape if they action surged, but they could only do that twice, and they probably wouldn't do much excess damage over wildshape in any case. The only way for a fighter to kill a moon druid in a 1v1 would be to win initiative, attack before wildshape got up, and do some pretty insane damage. I'm not even sure if it would be possible to do that sort of damage. A quick estimate shows that if all attacks hit, and assuming GWM and great weapon fighting style, would be 4 (2d6+5+10) = 4 (7+15) = 88, which becomes 2 x 88 = 176 with action surge. This is assuming average dice rolls, and that all dice hit. It's also technically ignoring the fighting style, since I'm not sure right off hand what rerolling ones and twos does to the 3.5 average on the d6. It will obviously raise it some. Especially with the minus 5 to hit, it's not very likely your attacks would all hit, but with pretty low druid armor class, it's a pretty good chance three of them would, and you might get a crit, or better than average damage, especially with the fighting style, so a fighter knocking a druid out turn one is certainly not impossible. For that matter, that's enough damage that knocking the druid out of wildshape and doing significant excess damage isn't out of the realm of possiblity, but you'd only have two chances to get it right and finish things. We're also not taking subclass features into account here.
A barbarian just doesn't have the damage output to do that, the way a fighter does, though.
What about paladin? Paladin has access to some really useful spells for this, being a half caster. They also smite up to 5d8, and with an auto d8 on their smite, that becomes 6d8. So, this would be an average of 27 damage added on a hit, with two attacks normally possible. So weapon damage +54+10. This would not be enough to break wild shape. With feats, it's certainly possible to, though. Polearm master or any feat that gave a bonus action attack, would move this to 1.5x as much nova damage. Great weapon master would add a +10. So with those two together, we'd get 2 (1d10+5+10+6d8) + (1d4+5+10+6d8) = 95 + 44 = 139 damage in one round, burning all three 40 level spell slots (but no level 5 ones). This also doesn't account for any spells we might use (like hold person), any abilities like channel divinities that let the paladin have advantage on attacks against one enemy, or (perhaps most importantly), the paladin capstones. I don't think Oath of Vengeance or Devotion would quite do it for us, but the Oath of Conquest capstone allows you to take an additional attack action as part of the attack action on your turn, which is pretty dang good. I still don't see a reasonable way to win here unless the paladin wins initiative, though. Most of the spells that would be good here (like hold person) are resisted by a wisdom save, which druids are good at. And you have limited smite slots (especially high level ones), that means you need to be doing enough damage quickly enough that you don't burn through your nova potential. It would be a heck of a fight and I think the paladin could at least touch the druid's actual non-beast health for sure.
So, to summarize, classes that can defeat moon druid:
Wizard. Full caster. Lots of tricks. I'd give the wizard majority depending on subclass.
Sorcerer. Full caster. Plus meta-magic as well. I'd call it 50-50, honestly.
Cleric. Not really sure. Full caster, but with a significantly less good spell list of tricks for this. I haven't played much cleric, though, and know they can be quite strong. My guess is, can win, but druid for the majority.
Monk. 100% can win. Iffy on if they actually WOULD win. Needs to be Way of the Open Hand. The druid has a lot of spell shennanigans they can just throw in the monk's way. And they can turn into a bird.
Bard. Can win, but I'd give majority to the druid. The bard gets less good spells for this in general, but also has magical secrets. They're also really really good counterspellers. And, depending on subclass, they can do stuff in this fight with their bardic inspiration.
Fighter. Possible to win, but would almost need to win initiative, and would also need to get really lucky rolls to hit and for damage. I don't think a fighter could beat a moon druid without assuming feats. Also pretty unlikely to get the win against a ninth level caster. The MUST TAKE OUT THE DRUID AND TAKE THEM OUT IMMEDIATELY!
Paladin. See fighter. Needs feats allowed. Also needs to win initiative and to have the dice be generally in their favor. I suppose they might win even with losing initiative, if they get a lucky hold person to stick or use some spell I'm not aware of, but it would be risky. Holy weapon would be strong for consistently pumped damage if they can't take the druid out turn one. Once again face the issue of dealing with a versatile full caster.
Ranger. Does have some spells, but really just can't win. Not without an extremely lucky save or suck spell.
Rogue. Doesn't have the nova damage output to win.
Barbarian. Doesn't have nearly the nova damage output to win.
Blood Hunter. Just no.
Artificer. I don't think so, but once again, I'm not super familiar with the class, and it will come down to spells to an extent.
Edit: Warlock. I skipped warlock. Initially I thought they had no chance, but on consideration, they do get Eldritch Smite (which is basically paladin smite on all 5th level slots), and also an invocation that adds charisma to their damage. Hexblade's curse also gives proficiency bonus to damage rolls and a crit on a 19 or 20. One of the invocations you can choose can also make your weapon a +3 weapon, so we can bypass the magic items rule.
So, assuming GWM, we have 2 (2d6+5+5+10+3+6 6d8) = 2(7+29+27) =126. With four level 5 slots, a warlock could do that twice. Or they could try to use a high level slot for something other than eldritch smite (maybe shadow blade or shadow of moil???). If the warlock loses initiative and has a way like polearm master to make bonus action attacks, they might wait and make three attacks instead of two on a turn, effectively make that 126 into a 189, if they did nothing else to buff it. If nothing else, that would significantly go above beast form hp and carry damage over the druid. Warlocks also have some pretty nasty mystic arcanums.
I think the three classes most likely to win are wizard, sorcerer and bard for sure. The least likely winners are the martial classes, minus monk which has a good chance, because like casters, they can also cheese.
