Thomas Jefferson (discussing offspring of white/black relationships) said that "three crosses clear the blood" (meaning that someone who had only one black ancestor after that many generations would count as white). Not quite sure what he meant but it could mean someone with one-eighth black ancestry. Of course in reality in America someone with one-eighth black ancestry is usually considered black unless some other identity supersedes that--Native American or Hispanic. (George Zimmerman seems to be at least one-eighth black, if his "Afro-Peruvian" great-grandfather was entirely of African ancestry.) Homer Plessy was one-eighth black but was arrested for sitting in the "white" cars in public transportation in Louisiana.
A great-great-great-grandmother could be pretty far back in time. One of mine was born about 1792 and died in 1852; another was married in 1770 (I don't know her year of birth or year of death).
“Do we even know if that great-great-great-grandmother was “full-blooded” Cherokee?”
Good point.