Hello, my name is***** and I have over 20 years of experience as a veterinarian.
I am sorry to hear about Johnny's cough.
If his cough is seeming to linger without any nasal or eye discharge a bacterial or viral infection seems very unlikely.
A dry cough can be a sign of heartworm disease. The most common symptoms for heartworm disease in a cat are coughing or gagging/vomiting. A simple blood test will help diagnose heartworm.
Another possibility, especially if he is allowed outdoors to hunt, is a lungworm infection. These can be more difficult to diagnose but checking several stool samples for eggs or trying a course of fenbenadazole (Panacur) worming medication may be diagnostic if he gets better.
Of course the other possibility is feline asthma. We can usually diagnose them with radiographs of the chest when they are showing symptoms. But if we don't catch them during an episode radiographs can look clear.
Heart disease is another possibility. Cats with a cough secondary to heart disease are usually very sick, very lethargic and refusing to eat. If he seems relatively normal otherwise, eating and drinking well, and active than heart disease is less likely.
Unfortunately a mass in his chest, trachea or larynx are also possible reasons for a cough.
It would be very reasonable to check a heartworm test as well as a stool sample and worm him with fenbendazole. If those aren't diagnostic and he isn't improving the next step would be to take neck and chest radiographs and look any signs of a mass and for changes consistent with asthma and compare them to the previous ones taken to see if there have been any changes if he has ever had radiographs taken before.
If no masses are seen and his radiographs look suspicious for asthma or even if they look clear when he isn't coughing then try a course of bronchodilators (oral like theophylline or inhaled) and inhaled cortisone (like fluticasone). In most cases inhaled formulation of both drugs work best. In some cases your veterinarian may start with an injection of steroids to get him some immediate relief. Inhaled formulations can work directly on the lungs, and thus tend to work better in the long run, and because they aren't systemic they have less side effects.
Rarely a nasopharyngeal polyp or a fungal infection can be the cause of a cough, so looking further with sedation and a nasal scope, as well as a look behind the soft palate may be needed, if he isn't coming along.
I understand that he has been treated with steroids, was that at all helpful?
If not then some testing would be a good idea to look further.
There aren't any good over the counter medications to give a cat with a cough, other than a wormer like fenbendazole, but hopefully I have given you the information you need to approach his cough and the diagnostic testing that will help pin the cause down and get him appropriate treatment.
Best of luck with your fellow, please let me know if you have any further questions.