At White's first turn there are $20$ possible moves: each pawn can move forward one or two spaces, or a knight can jump over the pawns to one of two positions each. Some moves are of course likelier than others (for example, in all the novice games I've seen, I've never seen anyone open with the king's pawn). Black also has $20$ possible moves at this stage.
At White's second turn, there is a small number of available moves based on what the first move was, plus a good player should take Black's first move into account. But what I want to know is, if we follow each of the $20$ scenarios created by the first move, count the number of available moves in each, no matter how strategically inadvisable the move may be (e.g., move the king forward) as long as it's valid by the rules of the game, delete duplicates (that is, count each move just once, not again and again in each branch that it's still possible to make it), how many moves are there?
Also, what is the most efficient way to tally up these moves? How far can we take this method, say, Black's second move, White's third, or further still?