I'm looking for the Holy Grail of multiplayer gaming

fall guys among us
Both infinitely replayable and has nothing to grind for except some cosmetics that doesn't affect game play

Battlefield 1

My interpretation of your request boils down to "what's a good co-op roguelike" where the grinding is the replaying.
So, depending on how many players you need it to support and preferred genres, you might check out games like
Risk of Rain 2
Enter the Gungeon
Children of Morta
Vampire Survivors
Streets of Rogue
Gunfire Reborn
Barony
There's also a game called Jumpship that i'm keeping an eye on the development of that's supposed to be hitting early access in the coming months.

There is a game i would recommend called Mechabellum. It's a mecha auto battler where you can't really move units between rounds. You build up a bug army trying to constantly counter your opponent with the 20ish available units.

Really depends on what you consider grinding.
Pretty much all MMOs or PVEs have you grinding for gear (helldivers 2 I don't feel is grindy in comparison, but some do)
Survival games like ark, valheim, etc.. Have you grinding for bases and the next section of the game
Pretty much all PvP games (CS2, valorant, apex, starcraft, Rocket league, etc...) have you grinding out muscle memory skills
The antithesis to these are instance-based games where at max you grind aesthetic gimmicks, but in single player games they don't have those like REPO where you always reset and fall guys where it is minigame based
The problem with these games is since you don't have a "reward for work" (grinding), people get bored of them.

SpiderHeck!

honestly check out archipelago, it's a framework that allows you to play a lot of different randomized games with your friends. you can play synchronously or asynchronously, and if you're handy with code, you can even add any game you want to it
- Thanks, that's interesting

Unreal Tournament 99.
- To this flavor: splitgate
- I was going for counter strike classic, but yup.
- I've spent countless hours in the 2004 version

Trackmania, although depending on how you want to slice it, you might consider it ONLY grinding.
Incredibly low skill floor (4 button racing sim) but with near infinite skill ceiling as you learn to master all the nuances of movement, surface types, tricks, etc.
Endless amounts of content with the seasonal campaigns, tracks of the day, and weekly shorts, but also just a full blown track editor for community content on the side. Each track is like a little puzzle where you memorize all the details then try and get your best performance. Play in an online server with your friends and just chat, listen to music, or watch a movie in the background. Find your favorite style and master it: tech, dirt, NASCAR, lol.
It's my favorite game to just turn my brain off and drive.

Sea of Thieves.
There has never and will never be any pay to win. Everyone has access to all weapons at any time, no unlocking, just pure skill.
Up to 4 players on your team (ship) open servers with other players all sailing around all the time. You can get in an organic fight over treasure, or matchmake for ranked battles.
All of the progression is cosmetic based.
The devs have been adding content constantly since launch that fleshes out the game systems and makes for more interesting interaction.
I come back to this game all the time. Highly recommend.
- This game is so much fun, even when I’m the loner getting my ass handed to me on my sloop (which is most of the time). Seeing a ship looming in the distance and wondering if it’s going to come after you is such a rush.
- Is it worth getting this late?
  
  Definitely, even better if you can try during a free weekend or get it half price on a sale.
  Amazing with friends, on your own or with randos.
  So many organic ways to interact and communicate with other players. Seriously cool, unique and worthwhile experience.
  
  Absolutely, it's still constantly getting updates, and got an actual anti-cheat recently which helped the health of the game a TON.
  Huge learning curve to the game, but there's nothing else like it.

Chess.
- -
✍︎ arscyni.cc: modernity ∝ nature.
- At the tightest-possible definition of multi-player
- lichess good
- Checks out

Tetris effect connected.
It made game of the year for a reason.
I think I've sunk 200 hours into tetris this year alone, I have it on like 4 systems including my SP handheld clone
Edit: my phrasing here was unclear, there effect connected has not been ported to portmaster or the ports collection, I use Tetris RR, a custom patched og GB tetris to have all the same amenities (hold, hard drop) as modern tetris.
If you like tetris and want tetris RR, find a GB tetris ROM and patch it yourself
A link to the Romhack (for the mods, this is a link to the patch, not the rom, i know VL has pirate content but this aint it, it does however have a link to their archive, where the rom does exist) https://vimm.net/romhacking/hacks/5813
- I got my hopes up it was on portmaster for a moment and you were talking about rg35xxsp.
  
  Oh, no I play Tetris Rosy Retrospective on that. All the new amenities on classic GB tetris.
  That's exactly the handheld I was referring to, the anbernic rg35xxsp

Fighting games:
Endlessly replayable because there's always more to learn.
No grinding because there's nothing to grind.
- I've enjoyed Tekken more than Mortal Kombat.
  I mostly played Tag Tournament 2

What do you want? Lots of games fit that description.
In some shooters, you basically only grind for cosmetics, like Overwatch, Marvel Rivals or Counter-Strike. Dota 2 is a Moba and also only has cosmetic unlocks, I think.
Maybe Minecraft or Terraria? I don't think you really grind in these games, but you do very similar stuff all the time.
- I consider digging hours for diamond as a form of grinding, and you have to get it for crafting advanced stuff. I could play creative mode, but at that point it's more like a 3d Paint than a game
  
  Vintage story has grind also but it makes it way more interesting and satisfying.
- These are the features that I want to avoid: grinding for resources, cosmetics, and anything that require repetitive and/or mindless action. Any genre is good, the question is more like a test to see if a game like this can exist
  
  No cosmetics pretty much disqualifies almost every game, because all of them have it, even old games.
  In my comment before, I probably wouldn't describe cosmetics as a grind. You just unlock stuff on the side, as a bonus, since most of it is random. It's not like you're going to play a thousand games of Ursa in Dota to unlock some rare skin.

The obvious answers are the games we endlessly replayed historically: Mario Kart, Goldeneye (VS mode), Halo (VS), Smash Bros.
If you specifically want ones on PC, I'd suggest Starcraft, Age of Empires, and probably Counter Strike (I wasn't into that one, but it had a huge following).
Many board games fit the bill as well. Codenames (physical or online at horsepaste.com) comes to mind, and another commenter also mentioned chess.
Basically any games that were made before endlessly grinding became a thing (yep, that's only been a thing for a decade or two).

In all honesty, Rainbow Six: Siege is as ungrindy as any game could be, and it is as endlessly replayable as there are combinations of all the active players. The whole game is about finding ways to use the deep sandbox to outsmart your opponent, utilizing yours and your teammates abilities in unique combinations and it’s wonderful

space station 14 (or space station 13 if you're nasty)
- There is not even the early access

Deep Rock Galactic!
- I've played it, but it has become a bit of a grind. The only reason for doing mission has become unlocking stuff
  
  I had issues with grind as well since the unlocks system required every single person to start at the same time and not play alone, as everyone had the same missions.
- Works just fine with anywhere from 1-4 players
  You can have a mix of friends and random players if you want a full team but don't have a group of four to play with
  The game is strictly cooperative so there's no incentive to screw each other over, fostering a generally good multiplayer culture
  There's lots of stuff to unlock but none of it is necessary for a fun and functional build
  Teamwork is encouraged and effective
  Missions are randomly generated but you've got years of additions to the random pools for lots of variety
  Absolutely no pay to win stuff
  There's cosmetic DLC, but there are still absolutely loads of great cosmetic options in the base game
  Edit: also, rock and stone
  
  For Rock and Stone!

The Finals or Rocket League.
- I also recommend The Finals! It's exclusively multiplayer, and the only "grind" one needs to worry about is unlocking all the weapons and gadgets; it doesn't take long to do, and you don't even need all of them if you aren't going to use them. But there's nothing to level up or upgrade, so once you buy an item, it's just a permanent part of your kit. There's no story or campaign that you have to progress, no cutscenes to sit through. You just launch the game, pick a mode, and queue for a match.
  I have like 750+ in-match hours logged, and have no plans on stopping.

Quake 3 Instagib.

Does Factorio count? It's a good game, you can play multiplayer, the factory can always grow (at least until your hardware, or in the extreme the software, can no longer handle it) and if you're grinding for something rather than automating it, you're doing it wrong.
- I guess it counts because you are not grinding, the factories are grinding for you. The problem is that it's a niche game and not many of my friends would play it

I play FF XIV online. There's as much or as little grinding as you want there to be. I'm in a really chill social FC.
- Can you advance in the game without any grinding?
  
  Yes, they give you the gear you need to make it each step of the way through the story.
  Crafting and gathering are optional. Relic weapons are optional.
  There are some players who like to role play. There are casuals, hardcore, and everything in between.

Old School RuneScape.
Some people will say it is grindy, but those people are just weak minded.
- People who imply rs has grind in it disgust me

Warthunder is really fun without much of a grind
- I especially love how warthunder js a f2p game but doesn't feel like it forces you into parting with youe money.
  
  Don't forget about the great bonuses for participating in the community forums, such as classified documents, toxicity, classified documents, and classified documents.
  
  This is how you bait a bot, folks.
  
  I agree :) and honestly paying for premium and planes you want is worth it. I have 750 hours and have spent maybe $100-200 on premium and planes. $3.75 per hour of birthday Monday seems decent to me. Better than my VR header which I’ve still only played down to $10 an hour