In Asimov's short story "The Last Answer," the Voice encourages Murray to find a way to end its existence.
Murray has just had a conversation with the Voice, and the Voice assures him it cannot be destroyed. Yet it sets Murray the task of trying to find a way to destroy it. Having this goal gives Murray contentment. Murray thinks the Voice wants to die:
For what could any Entity, conscious of eternal existence, want—but an end?
For what else had the Voice been searching for countless billions of years? And for what other reason had intelligence been created and certain specimens salvaged and put to work, but to aid in that great search? And Murray intended that it would be he, and he alone, who would succeed.
However, the above quote is merely Murray's interpretation of what the Voice wants. What we hear in the quote above is a voice—another voice!—inside Murray's head speaking, not the voice of an omniscient narrator authoritatively telling us what the Voice wants or the Voice itself stating its desires.
We can't know what the end goal of the Voice is. It is plausible that the Voice might want the peace of non-existence rather than the restless desires brought on by being immortal. However, it could very well be that Murray's quest to find a way to destroy it might simply be for the amusement of the Voice, who looks to others to keep itself entertained. The Voice may simply be manipulating Murray.
Murray is in the same dilemma as the Voice, for, as the Voice tells him, he cannot destroy himself. Whatever he does, the Voice can reconstitute him. Both now have to face the prospect of eternity. I would argue that one speculative way to interpret the story would be to see the godlike Voice as a compassionate entity that is giving Murray the intellectual stimulation he needs to cope with his new reality. Evidence of this is the story ending on the optimistic note of "with the thrill of purpose, Murray began to think." Of course, countering this is the dry last sentence "He had plenty of time," which raises the question of whether this task will occupy him for long enough—but maybe it will.
In the end, as noted earlier, it is impossible to know the final goal of the Voice.