The King is most like the Queen in the way he moves (rank, file and diagonal). A Queen in the centre of the board can move to 27 squares in a given move (considering no obstructions from any other piece). A Queen is worth 9 pawns. A King in the centre of the board can move to 8 squares in a given move. Therefore a King is worth 8/3 = 2.66 (recurring) pawns. Mathematically proven!
Although having said that, without the King, there is no game, so he's probably of infinite value. 
4 in the ending as in the ending it is usually clearly stronger than a knight if not a bishop
Maybe if you said instead, just for clarity; let's imagine that e.g. one of the bishops moved like a king. What would be its value?
This is probvably a better question.
Imagine a piece that moves like a King but is not a king.
This piece, for eg would be able to check the opposing king, or to be sacrificed. To me, this is what would make it a 'normal" piece.
Such a piece would clearly be stronger than a minor, because it would be able to mate with only the aid of the 'real' King, eg White king at f4, new piece at g4 and black king at h4. Whether such a mate could be forced, probably not, but there might be sacrificial combinations that lead to this.
So, I would guess between a minor and a rook.
I think the king is but a man, as I am; his ceremonies laid by, in his nakedness he appears but a man. Therefore when he sees reason of fears, as we do, his fears, out of doubt, be of the same relish as ours are.
--Henry V
just 2 .
What do you mean if the King were a normal piece? In the endgame it is very powerful. To equate it in the same manner as a knight or bishop at three points is very misleading.
If you had a King and a Knight and I had a King and a rook My king would be much more valuable than yours since it is invincible and I can still checkmate you. I guess it's all relative. If my King can assist in mating someone then it is worth more than a minor piece.
Well my king is also invincible. And I can still checkmate you.
rooperi, I think you can actually mate with King+fakeking. I can't make a diagram, but when I played against myself (of course, not a perfect defender) it wasn't too hard.
4
hmm one space queen, lets call it an ultra pawn!, id say its worth about 2 and 1/2 points :D
Less than the value of a pawn at the beginning of the game. I mean pawns can move 2 squares, the king only one.
At the end of the game, the fully passed pawn - 9.
At the end game, the king can be trumped by the rook along a file, or a rank but not both!
The bishop has no means of defeating the king.
The knight perhaps deals some sort of threat via forking.
Basically, in the endgame with more space, the kind a mobile piece in terms of direction is great piece. It can even dodge the hefty queen.
Defence - 10/10
As for attacking well...
Every piece can outrun it except perhaps a pawn - and what use is a piece in the end game if it's a miserable attackers? An example - in a rook and pawn endgame, you'd rather have an active rook than a passive one!
So 0.9/10 for attacking.
Now for the king v king. Completely equal, like opposite coloured bishops.
However, the king in combination with another piece can force capture of the opposing king.
Nonetheless, I think the king in the endgame is worth maybe again 0.9 since it cannot attack other pieces and so is wholly useless. The only purpose it serves is to act as cannon-fodder perhaps to create a diversion for a passed pawn.
However the king can act as a pawn in some ways.
At least the king has one special use - by castling it let's one rook get into the game (:
We are talking about a king as a normal piece not a normal king, which you can't put in check.
richie_and_oprah once told me the king was worth 4 points. That seemed like a reasonable value to me. The king's range may be short but it covers nine squares. That's more than a knight at its best. It can also stonewall another king much better than either a bishop or a knight. Its quite flexible and strong around pawn formations in an endgame. Yeah, 4 sounds good.
richie_and_oprah once told me the king was worth 4 points. That seemed like a reasonable value to me. The king's range may be short but it covers nine squares. That's more than a knight at its best. It can also stonewall another king much better than either a bishop or a knight. Its quite flexible and strong around pawn formations in an endgame. Yeah, 4 sounds good.
The king covers eight squares, at its best, just like a knight covers eight squares at its best. But the knight can get clear over to the other side of a board in 4 moves whereas it takes a king 7 moves. That has to count for something, if you're weighing for example how many squares a piece can potentially cover over time. So on the basis of these observations, the king would have to be worth less than a knight.
Actually though, the number of squares a knight on a random square can cover would be less than for a king, on average, because there is a smaller range of squares (in the center region of the board) from which a knight can cover eight squares. Short of the actual calculation though, and considering how fast a knight can move relative to the king, I think there's a case for saying they're exactly equal in value.
The KIng has no value and is invaluable, it can take other pieces but cannot be taken, it can move yet when it cannot move, under check, it is immobile, therefore dead....the discussion is academic..!!
It's like asking what is the value of the tin hat or the tin dog in Monopoly....no value but invaluable as you can't play without your movable marker....so no point to the game.
But a nice question nonetheless.
The question was posed as if we had a piece that could move like a king but not be the king.
Tonyfitz27, it is not a meaningless question. (However, a lot of people in this forum want to claim that no piece has any value, solely because the value of a piece can change based on the game situation.) But talking about the value of the king is not like talking about the value of the tin hat in monopoly. Let's try to be serious occasionally, people.
"but practically he never controls more than 6 squares"
FYI - its not any sort of valid point to just make a nonobvious assertion without any sort of stated justification.
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tonyfitz - The last part of your point about the tin hat in monopoly I missed, so I see your point, but as GoldenDog said, the point is the value of a king based solely on how it can move.
echece06 - OK, even then your point is wrong because the most squares the king can cover on the last rank is 5 (not 6). If someone is making a serious point (though btw I have not been involved in this discussion prior to #62 above) the least you can do is exercize care and consideration in your responses, maybe even read the thread title to make sure you're on the same page, (i.e. "If the king were a normal piece...").
NM Reb, you obviously like your bishops. But I kinda agree. Unlike a knight the king has no unusual movements allowing it to attack without being attacked. Its only strength is its flexibility, giving it power as a support piece or defender.
I've seen some people above kinda skirting around the idea of playing a 'piece elimination' chess game without a king, or playing an endgame without making use of your king, to try to gauge its value. I think of it this way - what would it be like to have 1 or 2 extra, expendable kings in addition to your main 'priceless' one? Would I see that as an advantage if they replaced the bishops, knights or rooks?
My vote is 3 points, 4 in the endgame. I'd exchange 1 or 2 knights, or 1 or 2 bishops for bonus kings, but not all 4 minor pieces. But I'd play to preserve the bonus kings for the endgames over bishops or knights. Unless doing so sacrificed a guarenteed bishop pair advantage...ah Chess, the game of exceptions.