So basically if I DM a game where the players don't level up and only find fixed equipment in dungeon rooms (a classic high level dungeon crawl) I didn't play a RPG? These kind of consequences just shows the absurdities that follow with the Codex obsession of trying to define a RPG by a fixed set of necessary and sufficient conditions or properties.
Well, you more or less need necessary and sufficient conditions to define a thing. Maybe we lack the necessary knowledge to define RPGs very well, but that doesn't mean there isn't a definition in first place.
At any rate, here is my take on this. You can't have an RPG, or a story, or much of anything, without progression. Now, this "progression" can be very abstract. I mean, even going "there" and then "back again" is a kind of progression. Of course, the thread is not about this abstract take on progression, but I wanted to mention this for a reason. When we consider tabletop RPGs, we do need some kind of progression. In particular, we need a character progression, that is, a progression of the character changing in some way in order to form the story of the game (in other words, we need an actual role that is played during the game).
This might seem like a story focused approach to RPGs, but it is actually easy to see how this applies even to D&D and other RPGS from that time. In D&D, your levels and classes represent a clear progression inside some kind of archetype from pulpy fantasy stories (the kind found on the so called Appendix N). Take a Fighter, and you have someone who goes from a tough, but very much mortal, warrior, to someone who can fight monsters alone, to a man who is an army by himself to a lord of soldiers who establishes his own law on the land. A thief goes from a street urchin kind of guy, to someone able to bypass all kinds of traps and rob others blind right under their noses, to a master of a crime syndicate.
The magic user goes from an apprentice capable of bringing about a single spell about each day (although, do note, 1st level spells can bring about a wide and interesting variety of effects, rather than just 1d6 of damage) to a pursuer of the arcane, capable of incinerating thousands of cubic feet, to an aspiring master of the inscrutable arts, with his own tower and apprentices, and capable of casting spells that blur the division between mortal and immortal. The cleric is a bit of an odd man out because, at least initially, the template was taken more from Christian tradition than anything, with spells that mirrored well known miracles. The cleric would start from a minor saint, going to become a beacon of light to those around and eventually becoming a great bishop with his own temple and performing feats such as resurrecting long dead people.
Someone might complain that this is not the point of progression in D&D. The point is not to change your character's position in the story, but rather to give you abilities to best the traps and whatnot of the dungeon. The point of gaining XP is not to become a great bishop that can perform miracles, but to be able to resurrect dead party members without paying heavy fees. That the point of becoming a fighter level 13 is to be able to withstand a lot of damage and manage to hit monsters with low armor classes. But I think it is important to understand those things together. You can't have a character with class mechanics that are disconnected from the game reality.
Anyway, although you can't have purely abstract mechanics in an RPG, that have no representation in reality (i.e. something that would be pure "crunch"), you can have things that have all sorts of things in a game without formal mechanics backing them up. For instance, depending on how the PCs converse with a ship captain NPC, the GM might allow them to hitch a ride on a boat without paying money for it, without a die roll to see if there is a problem during the voyage and even without rolling any kind of reaction roll; maybe based solely on the reputation of the PCs and how they approached the sea dog.
Thus, you can have character advancement in RPGs that is not dependent on any kind of hard mechanics. Jeepform is a mode of RPG where there isn't any kind of ability advancement (in fact, determining how competent your character is at something in jeepform is kinda besides the point). Also, many single-shot adventure games that are played on conventions, or just because you want to try something else, lack any kind of numerical advancement either. So, my point is that, at least in P&P, it is clear that this kind of advancement isn't needed.
Still,
Sigourn, you made it clear you want to talk about CRPGs. Sorry for the long detour, but I think understanding what could be done in a CRPG requires you to understand what you can do in P&P. CRPGs aren't the same thing as RPGs, of course. But they are single-player games that nevertheless try to give a context, a story and setting, for the mechanics and hopefully make them come a bit alive on the player's imagination. CRPGs are then games that are similar to pen and paper RPGs in some ways. This is not a real definition (which is perhaps what
moraes was driving at) but more of a subjective assesment. Who is tall and who is short depends on those who are around you, and being an RPG is a question of being similar enough to RPGs that the category makes sense.
In principle, you can do away with character progression based on hard rules in an RPG. You can do away with a lot of things. You could swap the combat system of the game for an abstract game where you roll dice until an outcome is decided. You could take away dialog and leave only some other mini-game to influcence NPC reaction towards you. You could take away all combat. You could reduce the exploration in the game and teleport the player around while mostly playing through "text adventures" with skill checks. You could make dungeon exploration into an abstract board game where you draw cards to determine what you find on each tile you land on. None of those things will absolutely not make your game an RPG much in the same way that no matter how short you are, you still have some tallness; even a drawing on a paper sheet has some tallness, the size of the thickness of the ink layer.
But like in the previous example, you would just be making your game less of an RPG, unless you are actually giving up these in favour of something else that would work well in its stead. The jeepform mentioned above does away with a lot of stuff because it is focused on dramatic situations, looking even a bit like a drama exercise. However, taking away mechanical advancement from CRPGs is just shooting one's foot, since the mechanical part is exactly where the computer shines, while the story crafting are only pale imitations of what you have in a P&P game. I mean, you can give attention to detail in a computer game you couldn't in a pen and paper one, to be sure, but you can't really give agency to the player in the same way. In fact, if you focus too much on making your story elaborate, it can end up being less of an RPG because the player can't really choose how to interact with it.
TLDR: If we are talking about CRPGs, there is little you can do that would make something not be an RPG. If you want, Doom, Zelda, Disco Elysium and even Tetris can be RPGs (where you play the role of a rocket engineer forever frustrated by soviet bad delivery practices and political enemies stealing your work). The issue is how much of an RPG your end game ends up as. Jagged Alliance 2 is more of an RPG than X-COM, since you have a clear in-game identity and since the units are much more human-like. Fallout is more an RPG than Underrail since the combat system in underrail is more abstract and with side-effects that make no sense. So if you want to remove progression from an RPG, but your game will probably be less of an RPG for that.