My son who is 1 year old has had a cough for over a month now?

Well to start it all off. a little over a month ago i took my son to the doctor for the cough and they told me nothing was wrong, it had gotten worse so i took him to the ER and they also told me nothing was wrong. a couple weeks later i took him to the doctor and they finally told me it was a upper respiratory infection and gave him amox. He took all the medicine but still had the cough, so i took him back to the doctor and they gave him a different antibiotic agmentin. He has finished his antibiotics and he still has the cough. im going crazy trying to figure out what could be wrong with my son. hes not acting like hes really sick, although he has been fussy but i dont know if hes getting some more teeth or what. i was hoping maybe someone could offer me some advice or maybe has went through the same situation. i also thought maybe he has allergies and they are not catching it. his doctor said its not ashma cause hes not weezing at all. he has had no fever or anything like that. just a stuffy nose and a cough that will not go away.