**EDIT**

Unfortunately, the details will not be ready until after the
bounty expires.  Someone else may be able to use the
sketch below.

Let S be the multiset of pieces used at the start of a chess
game.  Let Q=(64_S) be the number of possible arrangements,
one piece per square, using S on a chessboard.  Q is (the
value of) a multinomial coefficient using data from S, and
is about 4x10^42.

If one looks at the number of possible arrangements
using a legal submultiset S' of S (in particular the collection
of pieces arising from a legal game of chess involving no
promotions), one encounters an expression like
Q(1 + 30/33*(1 + 29/34*(1 + 28/35...))), which evaluates to
something less than 5Q.  Further, considering legal
collections T, one needs only those submultisets of T which
are legal and have not been considered earlier in a sequential enumeration
of legal submultisets.  Thus, a good approximation to the
total number of all positions considered should be the
sum over T a maximal legal multiset of terms (64_T).

The T I have found so far that gives a maximum value
for (64_T) is slighly less than 8Q, involves one pawn
capture and 3  queen promotions of different colors.
After two pieces being captured, it is hard for me to
see any optimal T occurring.

There are 165 possible multisets arising from promoting
8 pawns of one color.  I suspect an upper bound for
the number of maximal legal multisets is 165*165, given
that 2k promotions usually require k captures.  This
suggests an upper bound of 27225*8Q for the number
of all possible positions, or about 217800Q.

I was hoping to find a T such that I could beat Joel
Hamkins's ratio of legal game positions using T/(64_T)
while also satisfying (64_T) > Q, but time has run out
for that.  It looks like 2^-32 remains unverified at this
writing.

**TIDE**

If the idea is to come up with a rough figure, the following
approach using labeled pieces might help.

Consider the white king's pawn.  It cannot inhabit 24 of the
squares on the chessboard  (I assume promotion is strict
and monotone.  I let others argue whether to add 8 to the
number 24.) .  For the other king side pawns 24 grows to 26, 30,
and 36.  Starting with these ratios, a set of four pawns labeled
with their starting squares can occupy about 9% of all possible
positions available.  Raise this to the fourth power, and one
gets a ratio of 6x10^-5 as an upper bound on positions of
labeled pawns, where as a rough count I include pawns sharing
the same square.

Now the bishops provide another factor of about 6x10^-2,
the two kings together a factor of about 7/8, and the rest provide a factor
that is likely closer to 1.  This gives a weak upper bound for
all 32 labeled pieces of 3 x 10^-7 fraction of legal positions to all positions.

I imagine with refinements one can probably subtract one more
from the exponent.  Hopefully the fraction involving unlabeled pieces will not be much different.

I just noticed domotorp's comment regarding positions having just 28 pieces.  While this suggests the fraction
of legal positions may be higher, for the labeled case, the ratio (all possible for 28) to 
(all for 32) should still be small, and should have a small effect on the rough calculation.

And now I just noticed more recent comments from Joel David Hamkins, who made some
similar and more refined observations.  Perhaps he can place the above musings on a more
rigorous footing, and maybe someone can convert the labeled analysis to an unlabeled one.