Get it out of your head that odds matter when dealing with infinity.
Quote me some astronomical number of trials it will take for those monkeys to achieve their goal, even if just expressing the number itself fills up my hard drive, although I'd appreciate you not taking up all my space. Just to be on the safe side, square that number a few trillion times (although the whole universe is now out of hard drive space). We then assign that monstrous number that can't even be physically expressed to the variable N.
This is the kicker part, so read carefully. By the definition of infinity, for every number N, there exists N+1. The monkeys are finished.
Of course we're still dealing only with theoretical monkeys.