It doesn't have to be related to blood pressure or BP meds.
For instance, another common source of a chronic cough is acid reflux. It can be managed with meds, but often the result is a reduction in cough frequency, not eliminating it entirely.
My doc once told me: a cough is initiated from one of three directions: the stomach, the lungs, or the mouth/nose. Figuring out what direction the irritant is coming from is the first part of the puzzle.
*shrug* chronic cough is a thing for folks getting up in years. I've had it, and friends and family around me have as well. Some still do. Doesn't mean they have unmanaged health trouble.
For those who keep bringing it up every time they see Josh coughing, my recommendation would be to stop worrying about the man's health, unless you're his doc. And if you're his doc, what the hell are you doing on the interwebz, get back to work! Heh.
Go Vols!