Consider Neil Gaiman's story "The Goldfish Pool and Other Stories" (yes, that's the name of one story):
He nodded and grinned. "Ornamental carp. Brought here all the way from China."
We watched them swim around the little pool."I wonder if they get bored."
He shook his head. "My grandson, he's an ichthyologist, you know what that is?"
"Uh-huh. He says they only got a memory that's like thirty seconds long. So they swim around the pool, it's always a surprise to them, going 'I've never been here before.' They meet another fish they known for a hundred years, they say, 'Who are you, stranger?'"
The problem of immortal boredom solved: Just have a bad memory! Then even seemingly un-repeatable pleasures (meeting someone for the first time) become repeatable. Now you might say, wait, when I was thinking about immortality I wasn't thinking about forgetting everything and doing it again like a stupid goldfish. To this I answer: Weren't you? If you were imagining that you were continuing life as a human, you were imagining, presumably, that you had a finite brain capacity. And there's only so much memory you can fit into eighty billion neurons. So of course you're going to forget things, at some point almost everything, and things sufficiently well forgotten could presumably be experienced as fresh again. This is always what is going on with us anyway, to some extent. And this forgetting needn't involve any loss of personal identity, it seems: one's personality and some core memories could always stay the same. Immortality as an angel or transhuman super-intellect raises the same issues, as long as one's memory is finite. A new question arises perhaps more vividly now: Is repeating and forgetting the same types of experiences over and over again, infinitely, preferable to doing them once, or twenty times, or a googolplex times? The answer to that question isn't, I think, entirely clear (and maybe even faces metaphysical problems concerning the identity of indiscernibles). My guess, though, is that if you stopped one of the goldfish and said, "Do you want to keep going?", the fish would say, "Yes, this is totally cool, I wonder what's around the corner? Oh, hi, glad to meet you!" Maybe that's a consideration in favor. Alternatively, you might imagine an infinite memory. But how would that work? What would that be like? Would one become overwhelmed like Funes the Memorious? Would there be a workable search algorithm? Would there be some tagging system to distinguish each memory from infinitely many qualitatively identical other memories? Or maybe you were imagining retaining your humanity but somehow existing non-temporally? I find that even harder to conceive. To evaluate such possibilities, we need a better sense of the cognitive architecture of the immortal mind. Supposing goldfish-pool immortality would be desirable, would it be better to have, as it were, a large pool -- a wide diversity of experiences before forgetting -- or a small, more selective pool, perhaps one peak experience, repeated infinitely? Would it be better to have small, unremembered variations each time, or would detail-by-detail qualitative identity be just as good? I've started to lose my grip on what might ground such judgments. However, it's possible that technology will someday make this a matter of practical urgency. Suppose it turns out, someday, that people can "upload" into artificial environments in which our longevity vastly outruns our memorial capacity. What should be the size and shape of our pool? Update July 4, 2016:
See my new short story on this topic: Fish Dance (Clarkesworld #118, July 2016).