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I think some positive proportion, like 50% of all positions with legal pieces may arise in a legal game. I also think that the upper bound of $10^{47}$ in wiki is wrong. Consider only the 28-piece, no pawn configurations. If all pieces were different, this would give $(64!)/(36!)\approx 3.4\cdot 10^{47}$ positions. Also, we have $5$ ways depending on how many white/black pieces we have and we have four choices for each pawn promotion, giving an extra $4^{12}\approx 1.6\cdot 10^7$ multiplier. Of course now we have to divide with the figures that are the same, but that should be around (and here is the only place where I make an estimate) $4^8=6.5 \cdot 10^4$ usually, as there are 8 non-king pieces, on average 3 of each, and I rounded up because factorials are like that... So multiplying and dividing these numbers we get about $10^{50}$ positions. Of course because of the kings in chess (few because of too many bishops on same color) not all of these are possible, but we were quite generous when not counting positions with pawns, so I believe that there are at least this many legal positions that can arise during a game and the same order of magnitude when counting positions that can be set up with legal pieces.