Although I believe they eventually go on their own I got rid of some intransigent warts on my son's hands many years ago by the application of neat tea tree oil. Be warned that applied in this way it can cause some redness and a little soreness but they went fairly swiftly after application of the oil having been there months and months!
When i was a teenager (maybe a bit younger) I had warts and in frustration i removed them with sandpaper. I recall also biting them off. They never came back and I'm pretty sure it was the sandpapering that did it.
I had some small warts on my hand years ago, a friend told me to rub banana peel on them which I did on a more or less daily basis and they never turned in to full on warts and actually vanished completely eventually so this could also work.