This may seem far fetched but here is what happened to us. My son was like yours and just had this chronic cough, he didn't seem sick, it just never went away. Look out if he ever got a full blown cold or flu with it because then he would have croup to the extent that we ended up with an overnight hospital stay because he couldn't get enough oxygen. We tried inhalers, nebulizers, steroids, you name it.
The doc was at a loss, I was at a loss, my friends and family were at a loss and I was STRESSED OUT! Then one day my husband mentioned that he was sleeping on a down pillow and maybe he was just allergic to it. SERIOUSLY? Could it be that easy? Well, it was worth a try, within two days of removing the pillow, his cough was gone, and he seemed fine from then on. Crazy huh?
It has been a year and a half now and I must say, he does still have problems if he gets a cold. He will instantly get croup or start weezing and that is when the inhaler comes in handy, but never again did we have the chronic coughing or croup-like symptoms.
Long story to a short point, maybe his pillow or coats or something are down and bother him? Maybe you have an animal, a wool rug or anything else in the house that bothers him or aggravates his asthma symptoms? Who knows, but it is worth checking it out.
Good luck!