Here's how I look at this:
Every turn, the moves for one player can be represented in 21 base 4 digits, 20 of them for the joints and then a double boolean for the grip. Therefore, since it can be represented as such, it can also be represented as an integer between 0 and 4^21-1. For 2 players, you can represent it as 42 base 4 digits. For 3, 63. For 4 you have 84. Respectively, the integers are between 0 and 4^42-1, 0 and 4^63-1, and 0 and 4^84-1. Now, for every turn there is a set of these 42 digit numbers in base 4, so in two turns with 42 base 4 digits originally, you now have 84 base 4 digits. For 10 turns, you now have 420 base 4 digits. Therefore, you have 4^420 possible move combinations in 10 turns.
For reference, that's over 10^252 possible games. Assuming that we have 100000 super active players who play 1000 matches per day, each, against special bots who always use completely random moves and also play 1000 matches per day, which comes out to 36.5 billion matches per year, it would take us over 10^239 MILLENIA to have explored every move. That's also assuming we DON'T reuse ANY. Ever.
For reference, the universe is expected to last 10^100 years or so. That's a difference of 152 orders of magnitude. Alternatively, look at that as one number being multiplied with the other and 100 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000. That should be 152 zeroes. Feel free to count for verification, but even if there's not, add on a few more.
This is also for a single game mode.