Ok, here are some basics I can see. White is missing a pawn, a bishop, both knights, and one rook (the other being invisible), so 5 pieces missing. All 7 of whites pawns still alive have NOT made a capture.
Black is missing both bishops and a rook (3 pieces). Both blacks d- and e- pawns have made a capture.
Some additions: blacks a8 rook was captured by a knight, for certain. Whites black-squared bishop was also captured by a knight, as the black queen can't have moved.
Ok, my first post was incorrect. It is possible that white's a- or e- pawns have made two captures or zero captures, but no other amount.
I'm so in, but I've got to run right now. I hope this one is as difficult as the last one....
I was just asked for clarification as to whether White is in fact to move. It isn't provided with the question (simply the default for the position editor) so should not be necessary to solve.
Black's e and d pawns each captured king-side from their starting position.
And they didn't capture white's Knights.
I wonder how can we learn the position of white's rook.
If it was, say, on h5 before last white's move, it could be moved to a5 or b5 or c5 and we can't be certain about its position.
Even more, let's say that an answer is h5. Then the game might continue 1. Rg5 Ne7 2. Rf5 Ng8 - we have exactly the same position on the board with rook on f5 instead of h5. So if h5 can be an answer, then f5 can be too. I think this can help to rule out many of possible answers.
I think that we need to either prove that one of white rooks never moved at all; or that on white's last move it captured a particular black piece, position of which we can deduce; or it is on the square from which it can't make moves like I showed above.
It's on h7, and it was made from promoted pawn
Proof: as I wrote in #9, white's rook can't be on any square from which it can move to another square (from which it doesn't attack black's Q or K). The only variants are a8 and h7. A8 is impossible because it couldn't get there (neither a normal rook, nor a pawn to promote) without violating "royalty not moved" law. So, answer is h7.
White's h pawn moved to g file by capturing black's black-sq bishop, then captured blacks white-sq blishop on h7. Blacks g knight moved out and h rook protected the king, then white's pawn promoted to rook on h8.
Revenant's answer appears to make sense.
Maybe, revenant, but 1) you haven't actually proven that this answer must follow from the position, only htat the other ones don't work from the puzzle and 2) there couold be some reason why the other positions don't work. In short, there's a good chance you have the right answer, but you definitely don't have the right reasoning.
I'm going to make a post of everything we know so far, and will update later as we figure moer out.
Neither king nor queen has been attacked or moved.
From this, it follows that no opposing piece has been on either side's c or f file second rank, and only a knight on e or d second rank (will clarify this statement if necessary).Also, given the pawn structures, only a knight could have been on the c file, first rank, and only a kinght or bishop on the f file, first rank. (note: all of these are referring only to opposing pieces)
Black has, at some point, played dxe6 and exf6; neither captured piece can have been a knight. No other of black's pawns have moved (except obviously h7-h6)
Black is missing the a8 rook, which must have been captured either on a8, b8, or c8, but only by a knight.
Black is also missing both bishops, which must have been captured at some point.
White is missing his h-pawn, one rook, both knights, and his dark-squared bishop. This bishop must have been taken by a knight, which means that at least one of the black knights has moved.
The white h-pawn was the only candidate to promote, as it's the only pawn gone. If it promoted, it must have taken twice, in order to get past the black pawns. As the only two black pieces that could have been captured by the h-pawn (as the rook, as submitted, was taken by a knight) are the bishops, the dark-squared bishop was taken either on g3 or g5 and the light-squared bishop was taken on h7. Then the pawn promoted on h8, while the rook was somewhere else. Of course, it's easily possible that the pawn was just captured and didn't promote, but given the comments above and the nature of such problems, it seems likely that it did.
By the pawn structure, white has played c2-c3. White's a and e pawns started as a and e pawns. They either captured twice or not at all.
WanderingWinder, I agree that my reasoning isn't "pure" chess problem solution. Because it's based on the assumption that the puzzle has a unique answer, which is an argument "external" to chess problems (unless directly stated in the problem).
I'm 99% sure that the answer is right, but I too don't consider the problem solved.
WanderingWinder, correct me if I'm wrong-
Blacks d- and e- pawns MUST have captured white's rooks. Correct? (The knights are impossible and the bishop can't have moved, and we think the pawn promoted to a rook.)
The other option is that they captured a rook and a pawn, but that doesn't seem to prove much. Then we're left with one white rook flying around somewhere.
Not sure if that helps, I was just musing.
Continuing Nytik's thoughts.
If 2 rooks were captured, we already know what happened.
If a rook and a pawn. A rook was on e6, pawn on f6. But before black's pawn captured on f6, black's bl-sq bishop couldn't move out, so white's pawn didn't have enough "cannon fodder" to move to f file. So "rook and pawn captured" isn't plausible.
The black pawns MUST have captured Rooks. They could not have captured White's Bishop (which was captured on its home square) or either of White's Knights (which would have attacked Black Royalty). This leaves either the Rooks or the pawn which was captured on these squares.
The White pawn could not have been captured on e6 because it would have taken three Black pieces to get there, there are only 3 Black pieces missing, and the Black Rook could not have been captured by the h-pawn (as it was captured on a8, b8, or c8 by a Knight).
The White pawn could not have been captured on f6 because it would have to capture either the Black Rook or Black's dark-squared Bishop on f6. As noted before, the Black Rook was not captured on f6. Also, Black's Bishop could not have been captured on f6 because it would have had no way to get there since Black's e-pawn was still on e7 at the time.
Thus, the two White Rooks had to have been captured on e6 and f6. In order to still have a Rook on the board, then, the h-pawn MUST have promoted, and as noted in other people's analyses, the newly promoted Rook is currently on h7, with the pawn having captured first Black's dark-squared Bishop to get to the g-file, then Black's light-squared Bishop on h7 to get to a promotable position.
Very good, revenant... that means that it MUST have been two rooks captured... therefore the pawn WAS promoted... and you've already got the only way that can happen. Huzzah! 
Shame TheGrobe didn't get back in time, huh? This one was easier, anyway.
For this puzzle, one piece of knowledge is essential. In this game, no Royalty have been either moved or under attack. With that in mind, now know that there is a hidden White Rook somewhere on the board. Where is it located?