Viagra was a by-product/side-effect of a (ineffective) drug for angina and hypertension. I'm not sure how much UK government money was involved in that work, but I'm assuming a lot, as we're generally all for that kind of thing.

Also of relevance, the UK patent on using a chemical that gives you an erection as a cure for impotence was thrown out for being obvious. But the patents on how to mass manufacture it are still in effect.