At the end of series 9 episode 11, Heaven Sent,
the Doctor discovers that in order to escape the prison he's been put into, he must live through the same short period (it's not clear how long, but probably at most a few days) billions of times, resetting his body at the end of each iteration to the same state it started in.
So how old is he at the end of the episode? Clearly
his body has only aged at most a few days,
but what about his memories? Does he recall
all those billions of iterations of the same short period?
Is there any sense in which he's now
four and a half billion years old?