Well, I've been to the doctor. The cough was diagnosed as probably a virus which has been complicated by asthma - the virus itself has run its course by now, but my lungs are now irritated by all the coughing. Which means that I keep coughing and irritate them further, and so on. She says I may just have lungs which are easily irritated.
She's given me a preventative inhaler (which I 've never had before) and told me to rest. Which sounds good.
At first I was a little incredulous, since my asthma is generally triggered only by allergens and very cold dry air, and often I don't pick up my inhaler for months at a time. But now that I think about it, this happened to me on a lesser scale when I was 20: I had mild bronchitis, but the cough lingered on through most of a university term, causing me to miss a lot of football practice. And according to Mum, my little brother used to have similar problems.
The new inhaler does seem to have improved matters a little already; I'm still coughing, but not so breathless. I'm still quite sore, though. I don't think I'll be going running just yet. Maybe next week.