…Could a player Take Another Turn, and then, on their next turn, choose to Take Another Turn again, and again, and again, thus creating an infinite loop and breaking the game?

A: No, you are only allowed to take a maximum of two turns in a row using Take Another Turn. Just as in the case where Take Another Turn could be pulled out and used with Let’s Do That Again, you cannot get another “Another Turn.”
See also: What happens if Take Another Turn is played twice…?