Well, just because it’s bound to come up, there’s also the idea of quantum immortality: if the ‘many worlds’ interpretation of quantum mechanics is true, then whatever happens, as long as there is a nonzero chance of surviving, your subjective experience will always be of you surviving, as in all ‘branches’ where you don’t survive, there will be no subjective experience.
Consider the ‘quantum suicide’ experiment: you’ve got a gun in front of you, its firing mechanism keyed to some quantum event that gives it, say, a 50% chance of firing every ten seconds, such as the repeated measuring of the spin of an electron from an ensemble of electrons not in an eigenstate of the spin measurement. What happens after ten seconds will be that the world ‘splits in two’ (at least, on the most naive readings of many worlds): one in which the result of the measurement was spin up, the gun fired, and you died; and one in which the result was spin down, the gun didn’t fire, and you lived.
Of course, or so the argument goes, what you’ll experience then will be necessarily the branch in which you survive. But this is true about anything that could cause your death, and thus, your experience will always be that of survival, no matter how slim the chances. Thus, only situations in which your likelihood of dying is 100% flat—so-called ‘cul-de-sacs’—can conceivably kill you, and it’s claimed that no such situations exist. (It sounds like these cul-de-sacs are basically what you’re looking for.)
However, there’s a couple of problems with this argument: first of all, life or death rarely hinges on this sort of clear-cut quantum decision. Second, even in the simple example above, things are never as clean-cut as described above: the gun may misfire, the detector malfunction, an earthquake may destroy your lab, aliens may teleport you out at the last minute—all of which at some point becomes more likely than you experiencing spin-down after spin-down result—and so on.
But most importantly, dying is not very likely to be an all-or-nothing affair: even after the gun has fired, it’s not necessarily going to be lights out for you; you may be gravely wounded, but still, for the moment at least, have some remaining (and not likely to be very pleasant) experience. However, at that point, there may be no way out: this experience will gradually diminish, until it eventually fades completely—but there will be no sharp, all-or-nothing transition, and without such a transition, the argument doesn’t work.
This is a good thing: because while, if the argument worked, you’d be guaranteed immortality, in all likelihood, your experience isn’t going to be a pleasant one—you’ll be crippled, maimed, you’ll get sick, just as long as you manage to hold on to some thread of experience. As far as eternities go, this one will be rather more like hell than heaven.
But this probably doesn’t answer your question. Do cul-de-sacs, i.e. situations in which you are certain to die, exist? I think this is a very hard question to answer. In any kind of situation, it seems to me, you could come up with some kind of narrative that would allow your experience to continue in some fashion, if you’re sufficiently unfazed by its astronomical unlikelihood. You can always create swampman scenarios: you die, but somewhere, somehow, a bunch of molecules rearrange themselves by chance, producing a perfect copy of you, continuing your subjective existence.
Of course, at some point, the universe will end in some way; and for a sufficiently violent ending—a big crunch or a big rip, for example—it stands to reason that this will be the end of all life, as well. But in the case of a big freeze, for instance, while the energy available to do any kind of computation with—and thus, to run whatever hardware by then supports your consciousness—will become smaller and smaller, this merely means that the time between computational steps will increase ever further, but there won’t necessarily be a last one. (Indeed, Frank Tipler has speculated that in the case of a collapsing universe, since the processing capacity of such a universe would diverge, an infinite amount of computation will occur, including a perfect simulation of every conscious entity that’s ever lived; though let’s say that there’s a couple of steps in his reasoning that seem more than a little dubious to me.)