What is the best medicine for chronic cough problems?

Chronic cough. Chronic cough needs investigation. Most common reasons include allergies with post nasal drip, acid problems, tobacco use, heart problems-heart failure, and medications. The answer to your question is simple: treat the problem causing the chronic cough. For example, many people who suffer from chronic acid refulx might present with only cough. Treat the reflux and cough would resolve.
Depends... Treatment for chronic cough depends on the reason for the cough. Possible etiologies include bronchospasm from any cause, gerd, post-nasal drip or even a combination of these. All these diagnoses have entirely different treatments so it is necessary to have an evaluation by your doctor to determine the cause and treatment for your cough. Good luck!