It's an unsatisfactory answer to the greater question, but there just always seems to be a way to reword a sentence to avoid using this structure.
It can be by using different words to replace the passive verb, by making the sentence active, or by dropping the continuous where there's no significant loss of meaning.
Almost every hit in an Internet search for "has been being" is a page about the grammar itself, like this page here is. When you filter those out, you find unprepared speech, like someone posting a question about getting their phone fixed or a journalist quoting a source:
Hey my card has been being charged for apple internet for months and I never signed up.
Source: apple.com
Termite colony the size of Great Britain 'has been being built since the dawn of the Pyramids'
Source: Yahoo.com
Both these sentences could have been reworded:
Hey my card has been charged for apple internet for months and I never signed up. (removed continuous aspect)
Termite colony the size of Great Britain has been under construction since the dawn of the Pyramids (replaced passive verb with equivalent meaning)
Termites have been building colony the size of Great Britain since the dawn of the Pyramids (made the sentence active)