There is at least an in-game fiction of starting again. In truth it’s more of a Rashomon, the same story from different perspectives, and the restart is primarily an enabling mechanic, although those who’ve finished this masterpiece will know that it’s echoed beautifully through the looking glass by the final choice you make, making the whole thing a many-layered meditation on sacrifice and rebirth.

Definitely a masterclass in videogame storytelling, and one of the best (perhaps even, the very best) game soundtracks of all time.

While I agree that it's not really 3 times, I still think that it's useful to use that description when talking about games that do this, to be explicit that the "New game plus" option is integral to finishing the game.

I think this is an actual problem in Nier: Automata since (at least IMO and everyone I asked), the first ending is pretty underwhelming and it appears that the second playthrough will not change the story that much.

The game reaches "an end", but explicitely tells you to keep playing by starting a "new game" again. It's more like separate acts. The first two acts cover the same events, but the story is different because it's experienced through a different character each time.