Some say yes, you do need a hyphen because the term great is integral to the definition of great-grandmother. Otherwise, a great grandmother is just a grandmother who is terrific. Other people see no need and d not use hyphens for this.
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Great grandmother is a correct way to punctuate the way of genealogically identifying the grandmother of one's father or mother. This is due to a current tendency not to be heavy-handed in the use of dashes and hyphens, which traditionally are more costly in type-setting. The form great-grandmother nevertheless will be found too.
You add another "great" for each generation back in history. So the mother of your great great great grandmother is your great great great great grandmother.
You grandmother's father is your great grandfather and his wife (your grandmother's mother) is your great grandmother.
Your great grandmother's sister is your great-great-aunt.
She could be your grandmother. Or she could be your great aunt.