Can you use the term née twice when referring to someone married three times. For example, "My mother Jill Smith, née Brown, née White ..."
I do not think the term née should ever be used twice in the manner you described in your question.
For someone born to a Brown family, who later married a Green, then a Smith, then a Jones, I would say something like:
My mother Jill Jones, formerly Smith, before that Green, née Brown.
The term née means born, and a person can only be born once.
For registration purposes (in England/Wales) a maiden name does not mean the name a woman is born with, so the terms "née" and "maiden name" are not interchangeable.
In civil registration a maiden name is defined as "the name in which a woman first contracted a marriage". This will often be same as her name at birth, but not always.