Thesis Williams was right: an endless life would eventually become unbearable
Where it stands
Editors think about endings more than is healthy. A film that never ends is not a better film. It is a worse one, and after a while not a film at all.
Bernard Williams makes an argument like this about immortality, in his essay on the Makropulos case. In Čapek's play, Elina Makropulos has taken an elixir and lived for over three hundred years, and she has become cold and bored; she refuses to take the elixir again. Williams turns this into a dilemma. Either your character persists through endless time, in which case your projects and desires eventually run out and you face unending tedium. Or you change so completely that the later person isn't really you, in which case it isn't your immortality that anyone was promised.
I want to defend him. What makes a life shaped is that it is limited: choices matter because you can't make all of them. Remove the limit and nothing is ever urgent, nothing is ever lost, and the things that give a life its texture, risk, commitment, the sense of doing something now or never, quietly disappear.
That doesn't mean death is good. Williams himself held that an early death is a real misfortune, because our deepest desires give us reason to go on. It means the badness of death and the value of life might come from the same place, the limit, and you can't keep one and drop the other.
So: an endless life would become unbearable. Objections welcome, especially from traditions that have thought about endless existence longer than analytic philosophers have.