I had heard of fuller's earth before, but I had always thought it was named after someone named "Fuller." (Doesn't "Fuller's Earth" sound like the name of a science fiction novel?)
Anyway, so it turns out "fuller's earth" is called that because it was formerly used by fullers. What did they do with it? Why, they fulled, of course. And what's "fulling"? I'm glad you asked.
To "full" (MW10, entry 5, transitive verb entry) is "to shrink and thicken (woolen cloth) by moistening, heating, and pressing." You might guess it's called that because of something to do with filling the cloth in some way, but apparently not: this "full," which appears to be a homomorph of the more common word spelled the same way, derives from Latin "fullo" (by way of Anglo-French). (Whereas "full" in the sense of "having as much as will fit" comes ultimately from Greek plErEs. Sez MW10.)
I'm guessing that fuller is one of those now-obsolete professions (and presumably the surname "Fuller" comes from the profession), but I don't really know.
It turns out that fulling is also known as waulking in Scotland, where the songs waulkers sang while waulking are known as waulking songs. Not to be confused with walking songs, of course; such confusion would be fullish.