I'm not sure what the max is. You can have up to 6 in a party, up to 8 in each farm (not sure what the max on farms is), but you can also have monsters saved to a bank where they remain inactive.
My only concern is the 6 in the party. Any monster in your party (either the 3 active or 3 relief) gains EXP when you win a fight. However, the code for quick level-up is set so that the first 6 monsters you create in the game get the level boost (one level for each win). I'm looking to make it so that any monster in the party levels up quickly.