Actually there is no need for him to take the chunin exam, that whole ting is just to sort out the candidates... you have to remember that those who are watching (high level ninjas, kages) are the ones who decide who gets promoted and who doesn't. Remember that even if you win, even if you are the last one, that does not guarantee that you will become a chunin.
And now honestly do you see Naruto taking such an exam? Maybe only in base mode, and no rasenshuriken so that his enemies still have a chance xd