The main reason for my low rating is that i am not a very good chess player. Also I usually play in the evening before going to bed.
Keep in mind that there is no direct route from the number of possible board configurations (positions) to the number of games. If we assume that there are 100.000.000.000 possible positions (probably there are more), there are still only 20 possibilities for the first move and probably about 25 for the second move. this already makes 250.000 possible games after only two moves by each player and only one of those is called 'accepted queens gambit'.
Now for the number of games: It is relatively easy to calculate some number. the interesting part is to get a number you know something about. So In this case we would like to find an upper or lower bound for the number of possible games or at best get a very good estimate of the actual number of games. The easiest calculation would be to find out the average number of possible moves for any turn in a crazyhouse game. This number can easily be dominated by the pieces outside of the board, because they have at least 33 fields to be placed on (except for the pawns wich have at least 17). In contrast: A queen on an empty chess board has 27 or less fields to go to, a Rook has 14, a bishop has 13 or less, a knight has 8 or less, the king has 7 or less, and a pawn 3 or less (12 or less with conversion, 4 or less for ground row).
in practice the numbers for pieces on the board could easily be half or a third of what is stated above.
The other thing is the number of moves we want to look at.
@Ianalyse, you picked a number of 150 moves. I wonder if you ever played a game with 150 moves and wether it was interesting. Currently there are 8.843.053 games of crazyhouse, 7.380 took 150 moves or longer. Many of them end in one player having 17 Queens all on the board. Only 637 games were started from the Lobby. That is 0.01%. I've read somewhere that the average chess game takes about 40 moves. to me this sounds like a more interesting goal to aim at.
Now one way to get a number of reasonable games would be to look at 100.000 random games of crazyhouse from the lichess database and find the usual number of moves. now for another 100.000 games of this length (there should be at least 2.000.000 with a good choice of length) one would average the number of possible moves for each turn and player over all games.
This number potionciated by the average number of moves in a game should give you an idea of the number of games to expect.
Note: I left out some important factors here:
1) There are situaltions in wich a plyer has only yery few possibilities to move, for example if he is checked. there are also situations in wich a player has exceptionally many possibilities to move, for example if he has captured one of each type of piece of hit opponent and is not in check.
2) There are many more possibilities to place a rook and a pwan on the board than to place twi rooks, because they are indistinguishable; id does not matter which one of the rooks is placed on a particular field because they are the same.
3) This approach neclects very short and very long games.
4) this approach does not take the rule of 3 into account (the game ends if the same position is seen 3 times. so to get an idea of the theoretical maximum of games one needs a more sophisticated approach.
At the Moment I cannot tell you the numbers. One way to get them is to take a computer and use the lichess API. I am not too much into this, I would need to look up how this works.