I have used Dr. Scholl's plantar wart "Patches" - little adhesive dots with acid in them. My FIL, a podiatrist, recommends first trying the Duct tape remedy too, esp. if it is a child and surgical removal would be traumatic. If you're trying duct tape - it can take a LONG time - you have to be really patient and persistent. Also, although it's not really natural, Tagamet is used off-label as a wart eliminator. The little block dots - those are blood vessels. Warts are highly vascularized, that's why they bleed so much when you pick at them. To cut down on infection risk, wash (alcohol would be great) your tools and the skin well, before and after any digging.
I have a plantar that was surgically removed - it came back. FIL suggested that I be persistent about acid/trimming at dead flesh rather than try to surgically remove it again. Sometimes a slower removal results in a more complete removal.
Just one more note while I ramble on here - FIL (yes, we talk about warts a lot!) swears that he had a wart "cured" as a child by sticking it in stump water. (Standing water in a tree stump) It's worth a try!