1. At first glance, the game plays like a traditional roguelike. Random dungeons, turn based (but without separate battle screens), and permadeath (but see later points).

2. When you die, you reincarnate. You can choose a different class (which, let's suppose, can be a humanoid class or a monster class). When this happens, your new form will have higher stat growth, which further increases the more you reincarnate. You do start back at level 1, however, and the dungeon is re-generated.

3. Stats scale like they do in Disgaea. With enough levels and reincarnations, you can eventually get stats in the millions, and the game contains challenges appropriate for such levels of power.

Sounds cool! Would probably play for a bit.

No, not enough sexual inequality. :)

The closest thing I play like that is Rogue Legacy. Only it is a platformer.

You will certainly die but pass on the stats (fairy chest) and gold to your children and with you can buy items / level skills / increase stats. You have a few children to pick from, with different class and traits.

I have played Rogue Legacy, but the ideas I have for this hypothetical game are a bit different.

1. The game is turn-based and is not a platformer.
2. You create a new character instead of picking. In other words, you choose what you want to play as. (Maybe leveling up one class might benefit you when playing as a different class?)
3. Rogue Legacy's stats have effective limits, beyond which growth is slow and uncontrolled. For my idea, stat growth continues to scale much the way it does in the Disgaea series; in other words, you eventually get millions of stats, but so do the enemies in the most dangerous areas.

By the way, Rogue Legacy could have benefited from the ability to spend large amounts of money to customize the character you want to play as. Basically, after a while you have no use for the money you get, and only certain set-ups are viable, so being able to pay to use the setup you want would help solve both problems.

Nope, sounds quite dull.

I don't think that rewarding deaths is a good thing. It is counter-intuitive and goes against the rule "What doesn't kill us makes us stronger", subverting it entirely. But, well, I don't like the whole idea of permadeath either...

Sounds like a boring grind, honestly, and what would be stopping people from gaming the system by purposefully dying in order to get better stat growth? Random dungeons are also a serious turn-off for me.

words...
My days of grindy games are done.

Well, in Disgaea you get quite poor stat multipliers if you reincarnate at low levels.

One interesting idea I have is to remember the highest level the character has reached, and give bonuses based on how high that level is.

Another is to give permanent (even after reincarnation) stat boosts for each enemy killed after reaching max level. When you reincarnate, you keep the stats, but need to reach max level to get any more. (Also, perhaps stored levels would continue to accumulate at this point.)

These mechanics would reward players who manage to reach high levels in a single incarnation.

You know what would be interesting? If inside of the turn-based combat, you could decide on one of several ways of approaching combat (relying on speed to dodge or just soaking up things with defense, attacking every turn or using devastating attacks that take multiple turns to charge up) and have your character's stat boosts after reincarnation reflect those priorities so that your character/s gradually become more and more reflective of one's play style. And maybe better stat gains the higher the level you reach to incentivize playing as far as one can each time they're reincarnated.

I think a game like in the OP would need some kind of hook like that to stand out, honestly. There are a lot of grindy games out there, and a lot of games that rely on procedural generation, so without something unique it'd just be another face in the crowd that I could see myself ignoring. Then again, grindy games aren't exactly my favorites anymore, so I'm probably not the target market for such a game to begin with.

Wouldn't such kind of games be exploited by endlessly dying in order to boost the stats?

What's the selling point for the game?

You need to balance the big numbers anyway or it will not be fun 1 hit KO everything.
If you can achieve fun balance with 100 stats hero vs 200 stats enemy, what is the benefit of 10 million stats hero vs 20 million stats enemy?

Also turn base game by their nature eliminate the advantage of finger dexterity and makes combat straightforward. Through the combine strength of your stats, skills, equipment, items. either you are strong enough to win or you don't.
If you don't, load the last save point and go elsewhere to grind some more before attempting another pass on the boss.There is no need to die and go through the tedium of leveling up again.

What makes turn base games interesting is not stats, but choices you can make to change the flow of the battle. When to use skill of invulnerability, damage boost, status effect, characters placement, characters to sacrifice, enemy charm /blinded, reservoir of mana, enemy weakness, skills combo, environmental advantages, etc etc.

Increasing stats only just make me spam the same attack again and again to reach a certain point of the game where the enemy are strong enough to make me use my brain.

My kids enjoyed playing The Enchanted Cave quite a bit. In that game the stats are changed by the items you find, and there's a subset of items that persist if you decide to reincarnate (which you can do at any time before you die). There are also gems that increase stats, but the items were a lot more effective as an upgrade.

The good thing about it is that there's a limited number of floors (100) and when you reach certain levels you can start from that point instead of the beginning (it's optional, you can start at the top if you want). Otherwise it gets rather boring just going down to a place that matches your power, single shot killing enemies on the way and no items of interest to find.