Here is an argument that suggests that estimating this number, even to within many orders of magnitude, is intractable.
As an illustration, consider two sets of positions. A is the set of all positions in which white has 8 pawns, 2 bishops, and no queens, and black has the same. B is the set of all positions in which white has no pawns, 5 bishops and 5 queens, and the same for black. B is about 30 times bigger than A. We should expect that most positions have this character: boards crowded with powerful pieces as a result of many pawn promotions, including a lot of underpromotions.
A given position in B may or may not be reachable. It's pretty difficult to get that many powerful pieces on the board without causing a checkmate. If such a position is reachable, then watching it be developed on the board would probably resemble a chessboard history in which two amicable superpowers cooperate very carefully to allow one another the utmost possible peaceful development of their respective civilizations. Every time they approach the brink of a Cuban missile crisis, they unexpectedly find a clever way to avoid a premature end to the game.
I could imagine that no positions in B are reachable or that some significant fraction of them are. Getting the answer would require developing an entire theory for positions of type B, which would probably be as much work as developing a topic of practical chess theory such as bishop versus knight endings with pawns.
And B is only one of many similarly defined categories. Each of these lives on some huge brushy branch of a game tree, which may have many disconnected parts and whose parts may or may not be connected to the starting position of chess so that they're legally reachable. Each category or subcategory may require significant theorem-proving in order to determine its contribution to the total number of reachable positions. The total amount of analysis to be completed may be many orders of magnitude greater than the amount of analysis that has ever actually been done on practical chess.