OK, that was rather silly of me. in THAT diagram, Black STILL has no valid previous move.
If the thing discovering the check is a knight instead of a bishop, it could be done. But White has both Knights. It cannot be a promoted Knight because there is nowhere for White to promote - White's missing pawns have to be the G and H pawns, which do not have any path to promotion.
EDIT: Wait, I think I found it:
2nd:
I was looking at DavidMertz1's analyses and I was thinking how do I leave the Bishop on the board?
EDIT: further analyses:
You guys found a possible last sequence, great. But a legal game leading to that position with the light square bishop must be proven possible, of course.
You guys found a possible last sequence, great. But a legal game leading to that position with the light square bishop must be proven possible, of course.
Well, since you insist. We have proven your supposedly impossible to be possible. Now you need to explain why you thought it was impossible :)
Good job - also, I mixed up the positions by mistake, it is the other way around - first position is impossible and the second position is possible, oops 
Now, all that's left is to explain why the first position is impossible. Anyone up for it?
Yeah sorry about that, I just shouldn't give hints, I always mix things up 
here's another one, it's not so hard, but I wanted to make one that's drawn and my engine says it's won for black. If you think it's win for black you can post your analyses as well, really, it isn't that hard.
This is obviously drawn. White moves Kg2 - Kh1 in perpetuity. Black can take all of White's black colored pawns, but that won't help. Black's king cannot penetrate until a light colored pawn is gone, but that will never happen. Also, there is no way to trap White's king and thus force White to take one of Black's bishops, opening a path.
And keeping in the spirit of the thread, the position is possible. White captures 2 black knights and 2 black rooks with his pawns, Black underpromtes the A,C,E,and G pawns to bishops, all other pieces are thrown away however you like.
You're right DavidMertz1! This is possible because of the Bishops being the promoted pawns.
the first one is impossible because of the centre pawns:
hmm... does white have enough pieces for that? I was thinking it's kinda the same...
Here's another one:
and another one
and this one
and last
273 is impossible since White has all his material but Black has 2 c-pawns. The others look possible to me.
Well that isn't really true...
The key, of course, is, if it's White to move, what was Black's last move?
Everything on the 8th rank is trapped. (And that actually IS the 8th rank, that's listed, so the board isn't flipped or anything.)
fxe6 is impossible since the h5 pawn has no way to get there unless it's the F pawn. Similarily, h6-h5 is impossible.
So that leaves king moves. Since the king is trapped, the last move by White must have been getting Black into check. But the White rook is very limited by the king and rook next to it. O-O is impossible, as someone stated, because it implies the White king never moved, and then Black has no way to get its king to that part of the board.
Which leaves Kxa2. I frankly see no reason why the following is impossible: