How many legal but unreachable positions exist? Consider this legal chess position which can never be reached from the normal starting position. 

    [FEN "6kb/6p1/6K1/8/8/8/8/8 w - - 0 1"]


The Black bishop is placed on h8, but with the Black pawn on g7, there is no way the bishop could have actually reached h8. How many such positions exist in chess which are legal but unreachable? Is there any research on reachable/non-reachable positions? 

I found that endgame tablebases don't necessarily take this into consideration, but if the number of non-reachable positions is significantly large, it might possibly help minimize the size of endgame tablebases. 

Here's a screenshot from the online Nalimov tablebases. 



Now, in this unreachable position, I can add another piece like a knight on almost every square. 



I can add an additional piece, like a rook. 



This can go on and on and I can keep adding more pieces, but all those positions would be unreachable. We thus end up storing unnecessary positions in the tablebase and increasing its size. 

Of course, it's a good thing for tablebases to have these positions if we want to use them for chess variants like Chess960, but they are not necessary for the standard version of chess. It would be quite interesting to know how many such unreachable positions exist. 

(Addition of more relevant tags suggested)
The fraction of all six-man positions that are unreachable from the starting position is very, very small. Removing them would have a tiny effect on the size of a six-man database.

Here is an example of how rare these positions are. First of all, we can't get into one of these unreachable positions without a bishop and a pawn, so we already have a small subset of the number of total six-man positions. But okay, say that one side is lucky enough to have one pawn, one bishop, and a king. The bishop will have to be in one of its own corners (not the far corners!) and the pawn will have to be diagonally adjacent to it. So only 2 out of the 64 bishop positions could lead to a trapped bishop, and out of those, the pawn has to be in the one spot of its possible 48 positions that would trap the bishop. So even if you have one pawn and one bishop, which is pretty unlikely, the chances of randomly getting an unreachable position are only (2/64) * (1/48) = 1 in 1536.

Let's do one bishop, two pawns, and a king just for fun. If the bishop is in its own corner (2/64 chance), the chance of getting one of the two pawns in the diagonally adjacent square is 2/48 (2 of the 48 squares in ranks 2 through 7 have pawns in them). If the bishop is one of the other spots on the first rank (4/64 chance - c1 and f1 are definitely OK!), both pawns have to be in exactly the right spot (1/48 * 1/47) chance. When I do the math I get a chance of around 1.33% for getting an unreachable position even if you start with this very unlikely great combination of material for generating unreachable positions.

My conclusion, as first predicted in my comment above and now backed up by some math, is that these unreachable positions are an extremely small subset of the set of all six-man positions.