Due to a bit of a bug, Birthdays The Beginning has a nearly impossible Platinum Trophy but it does still work; info for every trophy is included here! Feel free to use the comments if you have a question or info to share.
2018-03-04: The trophy is confirmed possible thanks to Tanuzomi on twitter: It takes 10+ billion cube years or about 600 hours of gametime, however.
2017-06-23: Possibly not. The update notes claim Riding the Waves of Time to have been reduced to 3 billion cube years, but it is not. For now the platinum should be considered effectively impossible. It’s possible the unlock either doesn’t work or it may still be 100 billion cube years (requiring whole months of leaving your PS4 on at all times)