A non-royal king is estimated at 4 pawns: stronger than a bishop or a knight, weaker than a rook.
if the king was just another chess piece, what would it's value be?

It has some advantages compared to knights and bishops (in terms of being able to hit any square near it), but also some weaknesses compared to knights and bishops.
It's biggest weakness is its lack of mobility. It's basically a pawn that can move in any direction. But this makes it weak to long-range attacks. Both a knight and a bishop can attack it without being in danger, themselves ...
So I'd estimate it at being around 3 points in value.
A non-royal king can checkmate a king, unlike a bishop or a knight.
A non-royal king covers >= squares than a knight.
A non-royal king can move to all squares, unlike a bishop.

A non-royal king can checkmate a king, unlike a bishop or a knight.
A non-royal king covers >= squares than a knight.
A non-royal king can move to all squares, unlike a bishop.
- A non-royal king must move directly beside a piece to attack it - both a knight and a bishop can attack a piece from a distance.
- A bishop at one corner of the board can reach the opposite corner in one move. A non-royal king would take seven moves to accomplish the same task.
- A non-royal king can find its mobility restricted by adjacent pieces. A knight, by contrast, can hop over adjacent pieces.
I believe a non-royal king would be yet another minor piece, in value. Knight, bishop, non-royal king - they'd each have their own situations where they'd excel, or where they'd flounder ...

If the King is just as any other piece , we first need to identify the objective of the game before determining its value. So here placing the opponent's King in checkmate is certainly not the objective. Is it just capturing as many pieces ? If that's the case , the king would be somewhere between the pawn and the knight/bishop . Between 1-3. But I feel we might need to reconsider the values of other pieces as well. The King would be significantly better than the pawns and significantly worse than say a Bishop . And the range 1-3 would be really small to reflect that.
Pawn - 1
King - 2.5
Bishop/Knight - 4.5/4

That's an interesting question. But as Anonymous_Dragon pointed out, what would the new objective of the game be ? That can change a lot of things.

I'd say 2 points. Without the help of other pieces, it'll not be able to take on a bishop or knight, but two kings might be able to take on a knight. But since I believe two kings cannot trap a bishop unless the bishop is cornered, the bishop's value should be increased to 4.

I'd say 2 points. Without the help of other pieces, it'll not be able to take on a bishop or knight, but two kings might be able to take on a knight. But since I believe two kings cannot trap a bishop unless the bishop is cornered, the bishop's value should be increased to 4.
- I say that too
- Duplicate pieces = where the original piece can move
- You see which ones can move less space
- You can determine that by value of other piece
- KIng has no value but
- Queen is 9 points
Conclusion : King can move less space so it's 2 points
If the King is just as any other piece , we first need to identify the objective of the game before determining its value.
Why can the Pawn not be promoted to a Non-Royal King? The Crowned Pawn.
I don't see a description of that Chess variant on any website.
Also posted here:
https://www.chess.com/forum/view/general/pawn-promotion?page=2#comment-106199829

A non-royal king is like a stubbed queen, I think that how a real king can make a king and bishop, or king and knight, inssuficient mating material, means a non-royal king is like 4 points.
If the King is just as any other piece , we first need to identify the objective of the game before determining its value.
Why can the Pawn not be promoted to a Non-Royal King? The Crowned Pawn.
I just found this now:
https://www.chess.com/variants/chess
You can edit the rules, so the Pawn promotes to a (Non-Royal) King.
It's just called King though and looks exactly the same as a regular King piece. However, when you promote a Pawn to a King, the main 2 (Royal) King pictures both change and only those two can be checked (highlights the Royal King in red when in check).
As seen here:
I would have liked to see an option, in which you can choose between Non-Royal King, Queen, Rook, Bishop or Knight.
It currently shows: QRBN
Note: As the King is represented by the notation K, the Non-Royal King should have been NK.
Having a second (Royal) King as an option doesn't currently exist (notation would be K or RK). No variation/preset game has that at the moment on chess.com
You can't set your own notation either.
However, you can create your own custom variant which allows you to set 1 chess piece as a Royal:
https://www.chess.com/variants/custom
You can save the rules, so you don't have to repeat the input of custom rules every time you want to play the game.
Chess variants can also be played here: https://www.chesscraft.ca/design?id=5GH3
Edit: I didn't think of the idea of a 'Peasant' piece as a possible special piece (like a regular pawn, but no forward double move. Promotion to every other special piece except itself. So Non-Royal King too [Note: I don't think it should be able to promote to a Pawn, as Pawns only move forward {also not a special piece}. A backwards Pawn would be another special piece idea]). In hindsight, I think it's better than the Dragon as a special piece for a 9x9 Chess board.
You can create your own and share. (Unfortunately you can't filter search on their database, using keywords, to specifically find variants other people have created by searching the: [title/username/creation date] etc...)
There is currently no option to make the promoted peasant (new piece) a Non-Royal King. Which causes the game to have a fault, as the opposing King can be next to your newly promoted Peasant to King (shouldn't be possible, hence why the promoted piece should have been a Non-Royal King by default). So this variant allows there to be more than one royal piece on the board. Also allows more board size customisation.
also, i searched this exact topic and got this?
(cropped so much because of an ad below it)