General synonyms:
* Life story
* Life history
* Autobiography (if written by the subject)
* Memoir (if written by the subject, often focusing on a specific period or theme)
* Account
* Narrative
* Chronicle
More formal synonyms:
* Historiography (especially if focusing on historical figures)
* Biography (noun)
* Biographical sketch (a shorter version)
* Biographical study (a more in-depth analysis)
Less formal synonyms:
* Story of their life
* The tale of their life
* Their life journey
The best synonym to use depends on the specific situation and the tone you want to convey.