I've seen plenty of systems that the cost to raise the stat takes more 'points' the higher you go

hmmm, something specific right on the tip of my brain, but, yeah, plenty

There do exist JRPGs where stats don't change much over the course of the game. For example:

Final Fantasy 5 (FF5): Your stats change when you switch jobs and equip certain abilities, and some equipment changes your stats, but otherwise stats don't increase as you progress. (Instead, Level is a factor in most calculations.)

FF6: Stats only permanently increase if you level up with certain Espers equipped (a mechanic I find ugly because of missable stats), and the game is pretty easy even if you get no such stat increases. (It's worth noting that increasing stats can make you more powerful; if you have a level 99 character with 140 or so magic power (128 naturally + 12 from equipment) and you cast Ultima on a single target, the damage will overflow, causing you to do less than 1,000 damage. (Meanwhile, weak spells are doing 9999._

FF9: You get minor stat boosts by leveling up depending on your equipment (ugly just like in FF6).

Romancing SaGa 3: I haven't played this game, but apparently in this game a given character's stats are fixed (modulo equipment and at least one event that changes a stat for a specific character); instead, you get weapon and magic levels that increase. (SaGa Frontier 2 is sort of like that, except that the stats are absent entirely instead of
fixed.)

I always forget that that is the case with FF 5, but I think that it just shows how well made it was that you do always feel like you are progressing and getting more powerful when your stats aren't actually changing.

Well, the most important thing is, that I`m in control. I don`t like RPGs where the stats level automatically, like it is in JRPGs or the Baldurs Gate games.

Something in between Baldur's Gate and Dragon Quest. And it's not only the difference in power between the beginning and the end that spoils the immersion if it is just too big, it's also the sense of achievement. So if you make the levelling up too easy or if you want the whole game just about levelling up, it's boring. It's more like an action game then. CRPGs that I like most are about 1/2 story and puzzle solving and 1/2 levelling up and combat.

Funny that you mention the horror of minmaxing. Because for best results you had to time just right your leveling in the correct stats so that you have the maximum bonus...
My first rpg's were Fallout and Baldur's Gate so I kinda prefer that style, though later D&D editions have a better progression, with stats also increasing a bit and leveling feels a bit more satisfying from that point of view.

In open world games it makes perfectly sense that leveling up only makes a small difference, so that the game doesn't get too easy when you explore everything.

In linear games, I want a single level to make a huge difference already, though, so that I can notice my effort in grinding immediately.

Personally, I don't find the level-up stat growth of 3rd edition to be satisfying, as it favors focusing on the most important stat and ignoring everything else. You don't get enough level up stat points to afford to spend some to fix your weaknesses, and even if you did, it would still be better to focus on your main strength.
Interestingly, I prefer the reverse approach; I prefer it when you constantly get small boosts to stats, so that the improvement is gradual, and that you get some growth every playing session (perhaps even every battle).

Also, removing levels entirely makes it less likely for the game to have issues such as overfavoring overspecialization and missable stats.

While not perfect, I prefer it to 2nd edition where you could re-roll half a day until your roll had 18/00 Strength and decent average for example.

Well, it's hard to justify grinding in a linear game when the impact is too small. Say I get stuck at a boss. Then I want to be able to grind and come back and be able to defeat him. When I know leveling hardly makes a difference and I'd need to grind hours to be able to defeat him, then I'd usually just quit instead.

Removing customization also removes that problem. :p

My favorite growth system is learning by doing anyway. Especially if there's no upper limit, so that you don't get punished by going jack of all trades.

Are we talking videogame RPGs? Then stats are stupid, just give Basic and Advanced traits.

Stats in tabletop RPGs do one thing: they describe a broad class of tasks every character is supposed to be, individually, equally good at (okay it's an awkward sentence). Stats are needed because there's a countably infinite number of possible tasks in a tabletop game.

In a CRPG, the number of tasks is finite. And it is a crime against game design if a character is "strong" (say, STR == 19) but not strong enough and fails to e.g. move a crate (req. STR >= 20) because you decided to put two points in strength and one point in, idunno, ancient languages at a levelup a couple minutes ago, or because you failed a roll. There's a finite (and tiny) number of predetermined tasks, and the only strength levels that effectively exist are those defined by the tasks, like so:
STR=15: able to put on heavy armor
STR=20: able to move this fucking crate
STR=40: able to grab the end boss dragon by the tail and swing it around

So there might as well be a trait chain: Heavy Armor [+1] -- That Fucking Crate [+2] -- Tehpwnzor McAwesome [+3].

(Also 3E is best D&D.)

Actually, I was actually thinking of something like "learning by doing". Stat gains are typically minor in such a system, but they happen very frequently. If you spend enough time "grinding", your stats would, indeed, increase significantly; it's just that the stat gains don't all come at once.

I could, again, bring up Disgaea; in the post game, it isn't unusual for characters (at least those without capped levels) to level up after almost every kill (assuming the enemy is strong enough), but the stat bonuses from leveling up aren't lhat significant. The difference between level 2688 and 2689, for example, is very small. (Then again, in Disgaea's postgame, leveling up isn't the only, or even the best, way to improve your stats.) Then again, Disgaea's postgame *does* have an issue where most fights are either trivial or impossible.

Actually, the way it works out is typically more like this:
STR=15: Attacks do around 15 damage. An enemy with 100 HP will die in about 7 hits.
STR=20: Attacks do around 20 damage. An enemy with 100 HP will die in about 5 hits.
STR=40: Attacks do around 40 damage. An enemy with 100 HP will die in about 3 hits.

If you see it this way, then the stat is, indeed, relevant; it's not just a binary success or failure, and in-between stats have different behavior. (Interestingly enough, one reason for randomizing damage is to make it so that there is, for example, a difference between 34 and 49 STR in this example.)

Disgaea style for sure. I like the idea of being over powered but only if I want to work hard at it. I loathe short ceilings for these things.

Combat is a solvable math problem, and stats in particular are trivial. If Warriors deal increasing amounts of damage over the course of the game, run off Strength and max out Strength as level allows (eg. max stat = 15 + 3*level, 6 stat points per level so three go to Strength), just cut it out and have Warrior damage increasing directly with level.

Then, if there are two valid Warrior archetypes, e.g. one puts three points in Dexterity for mobility and the other three points in Intelligence for idunno bomb throwing, have these be traits Fast (can dodge attacks, move through battlefield control effects) and Gadgeteer (can cook and throw molotovs).