Bernard Williams has a famous paper, "The Makropulos Case: Reflections on the Tedium of Immortality." I've been studying the paper and here is my explication of the argument he offers for the thesis that immortality, if possible, would be undesirable:
[W1] A life is worth living if and only if it has categorical desires.
[W2] It is a non-contingent fact of any life we can call our own that it has a finite number of categorical desires.
[W3] In any immortal life we could call our own we would eventually fulfill all of our categorical desires.
[W4] So in an immortal life we could call our own we would eventually have no categorical desires.
[W5] Therefore an immortal life is not worth living.
A "categorical desire" is just a desire that makes live worth living. In this context, all we need to understand this sort of desire is just that it's the sort of desire that makes it feel, subjectively, like your life is worth living.
The argument is no good because [W3] is supposed to follow from [W2] but it doesn't. To see this, let's reflect on some St. Augustine.
Augustine may or may not have thought that it is a non-contingent fact about our lives that we have a finite number of categorical desires. The question would not have been very interesting to him, though. What matters for Augustine, so to speak anachronistically, is not the numberof our categorical desires but the nature of their objects. Maybe there is just one categorical desire and it cannot ever be fully fulfilled, given the nature of its object. So Augustine wouldn’t feel the need to dispute [W2]. But he’d deny that [W3] follows from [W2], and [W4] from [W3]. Granting we have a finite number of categorical desires, provided we had at least one which could never be fully satisfied, then, contra [W3], we wouldn’t eventually fulfill all categorical desires. And then, contra [W4], we wouldn’t face an indefinite remainder of our immortal lives having no categorical desires.
That Augustine thought there is at least one such categorical desire is no surprise: he thought it was God, who could always go on being enjoyed more because no one can have all of God at once and God is infinite. A lifetime of enjoyment of God is always, properly understood, a life we could call our own because God is infinitely and unqualifiedly good and all our desiring aims at the good in general.
So, if it is possible that God exists, then possibly you have just a finite number of categorical desires but one of these has God as its object. That one couldn't be satisfied, so you'd always have at least one categorical desire, so you're (immortal) life would go on (at least) feeling like it was worth living.