As I understand it, the treatment that works better than anything else in the studies is salicylic acid. You put it on, and then the next day you scrape it off with a knife until it bleeds, then put some more acid on it. It stings quite a bit, but unless it bleeds, it probably won't work. I think that when it hurts and bleeds the immune system starts taking the situation seriously and that helps eradicate the wart, you're also removing the infected skin of course.
If you do that, the wart should be gone in a few weeks.
In studies, salicylic acid works about 80% of the time. Other treatments like freezing don't work much better than chance.