Pathfinder 2e is a very good DnD-styled alternative that in my opinion solves quite a few issues that 5e currently has, namely the fact that martial classes get eclipsed by spellcasters, the overwhelming presence by dexterity as the best ability score there is and so on. Also, Paizo is not WotC! So you get to feel a bit better about supporting them.

I wrote a non-D&D TTRPG a while back, which I still run quite often: Whittledice and Parchment.

Whittledice & Parchment is a tiny, embeddable TTRPG – a game for your TTRPG characters to play, inside your existing game. Perfect for those times when someone doesn’t show up or your party just needs some downtime. If your current GM is interested in GMing two games at once, great – otherwise, pick someone else in the group.

Whittledice is print-and-play - just print out the rules, cut out the Character Cards, and get going.

I'm still working on it, but it's simple and definitely playable!

I've got another one that doesn't really fit with the others I posted, so I made a whole separate comment.

The Quiet Year is a map building roleplaying game. It has no combat; it is entirely collaborative storytelling. You play not as an individual character, but collectively as the spirit of the community. It takes a few hours to play a game, and it's relatively quiet and fun. We actually played it as a precursor to our current DnD campaign; we played the Quiet Year, and the resulting community that was made was the location for the first phase of our campaign, complete with issues that we had introduced, features we had sketched, projects we had seen started.

I am currently involved with 3 TTRPG games with 3 groups. One is straight DnD5. The second is Shadow of the Demon Lord, which is overall very similar to DnD mechanically.

The third is a hybrid mashup, and it's my favorite. It is using Blades in the Dark as the mechanical base, with the lore from a Microscope universe grafted over top of it.

Our group of 5 (later 6) spent about 6 months playing Microscope asynchronously using a Google doc (after an initial real-time session for our GM to introduce the system to us). We spent the time just building an interesting world, creating historical figures, historical events, etc.

At some point we decided there was enough there, and we picked a point in the timeline to have an adventure. We then used the Blades system to design characters and decide what kind of adventure to set out on.

The Blades system is far less crunchy than DnD and others. Instead of spending most of our sessions rolling dice to decide the minutia of combat, we spend the majority of our time discussing the sociopolitical ramifications of the really crazy decisions our characters are making. We have the freedom to retcon/alter/create anything about the universe we want, because we created the entire thing and made up the rules as we went. It is also the game I feel the most invested in, because I helped create it.

Disclosure: currently playing a 5E game, and the last game before that was a 3.5E. Here's an overview of games I've played in the past, but this is wildly out of date, and then a game I want to play on the future but I have literally zero experience with. So basically what I'm saying is that my list has no intrinsic value, but I enjoyed the question a lot.

Also at least one of these leans heavily into Wizards of the Coast, and I would also generally maintain that Wizards itself is actually not that bad, but Hasbro is pure fucking evil, which is a minor distinction, but an important one. It's the same stance that I used to justify playing WoW for a long time, and it's probably actually not that true; eventually the big wigs from the parent company infiltrates the top of the subsidiary company and turns it to shit too.

Played

7th Sea

A swashbuckling adventure game focused on, well, the seven seas. Pirates, buccaneers, privateers, sailors all kind of take the forefront. It's set in really-its-not-Europe-guys-no-really and you can be a variety of European nationalities factions. There's a bit of magic, a lot of swashbuckling, and a pretty fast and loose combat system. It's a bit of an adjustment from D&D and I actually am struggling to remember the core mechanics, other than that it is fast and loose! Theme A+, system B+, from what I recall

Vampire: the Masquerade (Storyteller System)

My inner goth theatre kid loved this, and I still love it. I would definitely play another White Wolf game as well. It is, honestly, teen oriented melodramatic bullshit, but tons of fun. If you wonder what I mean, read this overview of the clans and get a sense of the sort of game you'll be in. One of my favourite descriptions from the wiki:

clan fractured by madness, each member irrevocably suffering under the yoke of insanity.

That one sentence sums up the feel of the game. But if you were an early 20-something who loved Blade, The Matrix, Dark City, Interview With A Vampire, or even Buffy, then this was a way to play out something that felt similar to one of those movies. The Storyteller System is pretty good, and feels like it does focus more on the storytelling than pure combat, though combat was still fun and good.

Pathfinder 2E

I found Pathfinder to be the easiest drop-in replacement for D&D. There are things that I like better - the focus on feats which make characters feel a lot more distinct is probably my favourite. In D&D there are analogues to that, though - for example, my current character is a Warlock and the invocations have a similar feel to feats. Overall, P2E is a great system; easy to get into, fun to play, easy to make a switch to from D&D.

Custom Magic: the Gathering Dungeon Crawler

Okay, so this is probably my favourite. A number of years ago, my brother came up with a very cool and fun D&D / Magic mashup that I think Wizards of the Coast should just buy and release. You create a deck from a set of pre-defined cards, with some custom "class" cards that give you benefits, and then you run through a dungeon, with combat being played as a game of Magic, with the DM effectively being the Archenemy. We've only ever done one-shots, but they have been some of the best gaming sessions I've ever played. Here's his post on reddit about it.

Planning to Play

Numenera

I got a bunch of books from a humble bundle, and I just like the idea of the game. It is set a billion years in the future, and the rise and fall of civilization has happened several times. There is technology that is effectively magic - the Clark quote, "any sufficiently advanced technology is indistinguishable from magic" is basically the defining characteristic of the setting. The system itself seems pretty simple and easy to get into, and the character creation seems fun, and is about as far as I've really gotten into figuring out the system (I'm angling to get my D&D group to play this after our current D&D campaign ends... in about 2025). Your character has a sentence that describes what is effectively your "class":

I am [Name], a [Descriptor] [Type] that [Focus].

• Type - a basic class, of which there are three - Glaive (warrior-ish), Jack (thief-ish), Nano (mage-ish)
• Descriptor - an adjective that describes your class. The list is extensive, but are things like "Slick" or "Intelligent" or "Mystical" and these change how your class functions. "Cautious" for example, will increase your stealth and your perception, but decrease your initiative in battle.
• Focus - even more specialization. You might take "Brandishes a Silver Tongue" and improve charisma based things

So your character sheet at the top instead of something like "Jarvis, level 12 Bard" would be "I am Jarvis, a Learned Jack that Brandishes a Silver Tongue". I don't really know how well this works, but it seems like fun in theory.

BATTLETECH

The greatest tabletop spreadsheet simulator ever created. An interstellar warfare simulator largely carried out on planets and moons with centuries old stomp-y robots built on forgotten technology, lore-rich factions, and semi-hard sci-fi physics to tie it all together.

Recent resurgence is entirely due to the success of the HBS PC game which was piggybacked upon to release new books, models, etc. I haven't played the TT game in close to 15 years now because the player base had dwindled to just pockets of old people like me that aren't locally concentrated enough to actually get a regular game together. That is a possibility now as there are games at game stores and LFG now, but I haven't made the time to go and know that if I do so it'll just be another hobby I pick back up, which will inevitably result in spending inordinate amounts of money on boards, terrain, accessories, a resin printer, and the services of a miniature painter. So for now I play MegaMek when the BT itch needs scratching.

The game is fun, it has the right amount of logistical management in full campaigns without actually weighing you down with unnecessary minutia like some games and a casual scenario can be tossed together easily if you aren't in the mood for full campaign which is often weaved into someone else's campaign as you're largely a mercenary that can take jobs as you please. No magic, no aliens, no interdimensional beings, just humans doing human stuff a millennia+ into the future. Factions range from relatively "normal" groups to hyper-fanatical dirty clanners so you can play whatever part you want, but also suits people that just want to be themselves and do their thing while blowing up each other.

I've been in a Dungeon World game as a player for a while now and it's been a pretty interesting D&D alternative to try. I'm not sure I like it more than I do D&D – it's a bit too simplified for my tastes and the lack of critical successes/failures take away too many interesting moments to me – but it does have some noteworthy advantages.

I think the biggest of them is the fact that the combat system is much faster-paced. Instead of an initiative system, players just ask the DM when they want to do something. They roll for it, and if they fail badly enough, the DM "makes a move" against them. In practice, this usually means a player tries to attack something, misses, and as a result the monster hits them instead.

On the downside, this means enemies get almost no rolls whatsoever, and so they almost can't fail,¹ and I find the lack of, for example, a monster failing a stat-save to not get knocked off a cliff after being hit to be a shame. On the upside, this feels way faster and lets players act far more often, as well as dramatically simplifying things.

Overall, I feel like this change in particular is on the whole beneficial, as one of my biggest peeves as a D&D player so far has been that I feel like I spend a lot of time waiting to get to do things, and this solves that. Still, I can't shake the feeling I'd be having more fun if more granularity were available.

_{1. What more, if players get lucky, monsters might not get to do anything at all.}

Lancer. 11,000 years in the future humanity has finally gotten its shit together and is working to build a galactic utopia. This is hampered by the tyrants and corprostates that want to rule their own little fiefdoms instead of letting people be free. Player characters resolve problems by fighting in giant mechs. Also there are AIs from outside of causality; some of them are friendly but all of them are deeply weird.

Notable for having surprisingly excellent companion software.

Stars Without Number. I'm sure that I don't need to explain to a Kevin Crawford fan why, but...

I love sandbox. I love putting the power in the player's hands. The Firefly-like fun of "point your ship this way and see if you can survive on your wits" is incredibly satisfying, and the game system running it all is wonderfully simple.

I play (well, the campaign is on hiatus now, but our last session was not that long ago) In Nomine Satanis / Magna Veritas (aka INS/MV), that French TTRPG about angels and demons. I've written more extensively about my experience here on Tilde, but the short version is:

• Mechanically it's close to GURPS (later edition of In Nomine (the english version uses GURPS). Action resolution is done by throwing three d6, look at the first two for success/failure and look at the third one for the magnitude of said success/failure. The success/failure threshold can be adjusted by spending skill point and with contextual modifiers at the DM's will. For flavor, we call throw a d666 throw. Also for flavor, something satanic happens when the result is a 666, and something divine happens when the result is 111 (usually a flashy critical effect; "normal" critical happens when rolling a double).
• Story-wise it's contemporary satire of the French society. There's a huge conspiration ("the Great Game") that hides the fact to us normal human that everything in the Bible is real and that angel and demon are waging a secret war since Jesus' death. But being an angel doesn't means being nice (racist bigots are the backbone of God's army), and being a demon is more often than not more focused on corruption rather than senseless killing. And, this being a french game, there's a fair amount of internal political bickering and bureaucracy on both sides. Flavor example: you don't get experience points, but rather "administration point" that allows you to maybe get the new power/upgrade you want after submitting an official form. Those points are not gained naturally, but given as a reward by your hierarchical superior when reading your mission report.
• The DM is free to put whatever ambiance he/she wants, but the official adventures varies between epic (we've witnessed the birth of a new Demon-Prince with the magic of voodoo...) and humorous (... we had to communicate with her from within the dreamscape; being the mental equivalent of a 8 years old kid, it was filled with lethal princess, legally-not-jedis and super-demonic-stenai).

I will always evangelize for Barbarians of Lemuria. It's the first ttrpg I've liked enough to actually buy a paper copy of. It's incredibly flexible and incredibly simple and all of the roles are 2d6 sometimes with an advantage or disadvantage and you have to beat nine. That's it. You add direct numbers from your stat sheet and just getting 1 extra attack feels powerful. It's an homage to the sword and sourcery of old with heros that spend all their money at the end of each adventure. It's a good fun low crunch easy maths system with simple character sheets and minimal prep needed.

Through the Breach, a card-based TTRPG where you flip a deck of cards and can cheat a card from your personal deck if you don't like the result, assuming you have cards available. It's set in the Malifaux world, an early 1910s alternate timeline where a rift opened and introduced magic and another world into our world. It's got pretty much everything but leans more steampunk than anything else I think.

13^th Age (SRD) is fantastic. It's a D&D-esque system that is lighter on rules (especially coming from 4th Edition) and heavier on trusting the GM and players to make smart decisions. The coolest thing in my opinion is the way that they handle skills. Instead of being proficient in "Sleight of Hand" or Picking Locks, a rogue might put background points into "Cat Burglar" and then when asked to roll a skill check the player will ask if Cat Burglar applies or not. This can get interesting with more creative backgrounds. One of my favorites was a player who put several points into "Good Dancer" which he then successfully used for things that required light feet some types of social interactions.

Many of them, actually.

• Mythras - I've been pretty heavy into this for about 9 years now. Better combat, more flexible magic, culture and career (no classes) make mechanical differences. Feels like where I should be
• 13th age - really hit well with my group. we pull Backgrounds from it into any other games we can come up with.
• Cortex Prime - I have been playing this since Leverage, Firefly, and Marvel heroic were things. This is my narrative game. I think it does genre focused media based games better than anything else