Just so you know: Tipler's Omega Point scenario is the time reverse of a big bang expansion from a BKL singularity. The collapsing universe, filled with a plasma too hot and dense for any bound object to survive, is supposed to undergo an infinite series of "Kasner oscillations" which alternately squeeze the plasma from different cosmic directions, providing the energy for computation.
The scenario is very problematic. The plasma description will not be valid at arbitrarily high temperatures. Eventually the particles will be colliding so hard that they become micro black holes; some other dynamical regime will take over. Tipler has worked hard to contrive ways around this, but it's just really unlikely that an infinite sequence of Kasner epochs can be made to happen; especially, I would think, if you work within string theory, which behaves differently from field theory at high energies.
There is no consensus in string theory regarding cosmological initial and final conditions. String theory sometimes "resolves" singularities, i.e. provides a non-singular description of an apparently singular geometry (e.g. the "fuzzball" description of the black hole interior). However, there is no consensus on whether a big-crunch singularity will generically resolve and lead to a big bounce (as in "ekpyrotic" and "pre-big-bang" models), or whether it is simply the end, even in string theory.
At the other end, there is no particular consensus about the combination of chaotic inflation and the string theory landscape being the right way to think about cosmology. (I should probably emphasize that most "string cosmology" is actually about events in a single expanding universe - e.g. studying how the inevitable extra heavy particles affect measurable aspects of cosmic evolution like dark matter and atomic abundances - and not this mind-of-God stuff.) Its chief champion is Leonard Susskind, who is very eminent but does not speak for all his equally eminent colleagues. But let us assume this framework for the purpose of discussion.
Inflation is a hypothetical period of exponentially rapid expansion in the very early universe. In a field theory model of inflation, you start with a "scalar" field in a high energy density state, it dynamically relaxes into a lower energy state, and then inflation ends, being replaced by cosmic expansion at ordinary rates. For inflation to occur, the scalar field only has to have a few properties and so there are endless specific field theories which will exhibit inflation. In string theory, (see bottom of page 6 here), there are also many ways to achieve inflation.
In "eternal inflation", most of the universe always remains in the energy-dense inflating state. The relaxation into slower expansion only occurs in small, disconnected spatial regions, outside of which exponential inflation continues forever. In "chaotic inflation", the relaxation process sees the inflationary fields settling into different stable states in different regions. Maybe I should explain what a "stable state" is. In particle physics theories, particles usually get their mass by interacting with a Higgs field or fields with a "nonzero vacuum expectation value". The Higgs fields interact with each other and settle into some lowest-energy equilibrium determined by the form of the interaction (which can be quite complicated). There can be more than one such equilibrium.
In string theory many apparently stable configurations of the extra dimensions have been constructed. So in stringy eternal inflation, you suppose that different string geometries are being realized in different isolated regions of an otherwise uniformly and eternally inflating universe. Usually this is brought up in the context of anthropic reasoning; the hope is to predict the features of our local physics anthropically, since we can't be living in a region hostile to life.
Now we can think about this question of whether an infinite computation might get to occur somewhere in such a universe. The two standard cosmological scenarios for eternal life are the Tipler and Dyson scenarios. I've already mentioned that Tipler's scenario is dubious. Dyson's scenario is for an eternally expanding universe; something about stable islands of matter communicating with each other ever more rarely and weakly, with these interactions spaced out in such a way that they manage an infinite sequence of such interactions on a finite energy budget. If you believe in a cyclic cosmology, you might add to these scenarios one in which life persists through the bounce from collapse to expansion, but no-one has proposed a model of that.
I have not deeply surveyed the literature on eternal inflation, but I don't remember ever seeing anyone talk about one of those non-inflating regions entirely ceasing to expand and undergoing collapse. My grasp of the concepts is weak enough that I can't even say if there's some principled reason for this, though inflation is such a generic phenomenon, I would think that there must be models where a local big crunch can occur.
In the string theory context, people really started talking about a landscape in theory-space of many possible geometries, after the observational discovery of dark energy in 1998. That was at first difficult to incorporate into string theory, and the way it was achieved (in "KKLT vacua") involved the discovery of a new, very large class of stable string geometries. A universe with dark energy is one that expands forever, even at an accelerating rate (just not as fast as inflation). So if we suppose, as Susskind seems to do, that the landscape is dominated by these vacua, then it's the Dyson scenario, appropriate for an open universe, which is the relevant model of infinite computation.
Now here I am really overreaching what I know, but in discussions of these vacua - which have a de Sitter geometry - I often see it stated that in the end every particle ends up isolated from every other, alone in its own Hubble volume. So the Dyson scenario may require a flat universe, and may be impossible in de Sitter space. I think there's actually a paper saying as much. I'm not clear on this, but I don't think this geometry requires that literally every particle ends up in its own patch of expanding space. Just as a galaxy doesn't experience cosmic expansion, that only happens out in deep intergalactic space where the geometry is FRW, I don't see why a gravitationally bound system much larger than a single particle couldn't become one of these islands in de Sitter space (in which case, maybe you could hope for infinite computation, but not infinitely many states - you would end up repeating). It may only be the "big rip" scenario, in which the dark energy grows, that tears all bound systems apart. But I'm really not sure!
Really, I think these discussions about what's going on beyond our cosmological horizon are a lot like the discussions of the Fermi paradox. They are exercises in reasoning almost totally unconstrained by empirical data. String theory is supposed to be this unique mathematical structure and so you might hope that it simply tells you how string cosmology is supposed to be. But it's a work in progress, and in fact the cosmological question may be the same as the other big unresolved question, how to think about all those different geometries. Usually you just pick a geometry and study how the strings behave in it. You allow for some back-reaction, so the geometry adjusts to what the strings are doing, and in some cases you can even describe how one geometry becomes another (Brian Greene worked on this). But a conceptually unified approach regarding the whole "moduli space" of possible geometries is lacking. Other theorists like Cumrun Vafa and Tom Banks have approaches very different to Susskind's. (Vafa appears to be looking for a single preferred geometry, by using the Hartle-Hawking wavefunction, while Banks thinks moduli space is divided up and the vacua form disjoint groups that aren't dynamically connected.)
Final message: as currently described, the string landscape plus inflation is not generally thought of as allowing infinite computation or eternal life. But that whole cosmological conception may be faulty.
This comment is most informative, thanks.
In recent times, science and philosophy have uncovered evidence that there is something very seriously weird about the universe and our place in it. We used to think that there was one planet earth, inside a universe that is very large (at least 10^26 meters in diameter) but that the reachable universe (future light-cone in the terminology of special relativity, or causal future in the terminology of GR) was finite. Anything outside the reachable universe is irrelevant, since we can't affect it.
However, cosmologists went on to study the process that probably created the universe, known as inflation. Inflation solves a number of mysteries in cosmology, including the flatness problem. The process of inflation seems to create an infinite number of mini-universes, or "inflationary bubbles" - this is known as chaotic inflation theory. The physical parameters and initial conditions of these bubbles are determined randomly, so every possible set of particle masses, force strengths, etc is realized. To quote from this piece by Alan Guth:
The role of eternal inflation in scientific thinking, however, was greatly boosted by the realization that string theory has no preferred vacuum, but instead has perhaps 101000 metastable vacuum-like states. Eternal inflation then has potentially a direct impact on fundamental physics, since it can provide a mechanism to populate the landscape of string vacua. While all of these vacua are described by the same fundamental string theory, the apparent laws of physics at low energies could differ dramatically from one vacuum to another.
To top this off, the dominant theory about the spacetime manifold we live on is that it is infinitely large in all directions. If you look at this picture of a reconstruction of the large-scale structure of the universe, the idea that we are living in something like an infinite volume with a finite speed-limit and a uniform random distribution of matter and energy that clumps over time becomes plausible.
A final step along this line of increasingly large Big Worlds is modal realism, the idea that all possible worlds exist. Max Tegmark has formalized this as the Mathematical Universe Hypothesis: All structures that exist mathematically also exist physically.
If any of these theories turn out to be true, then we are living in a Big World, a cosmology where every finite collection of atoms, including you, is instantiated infinitely many times, perhaps by the same physical processes that created us here on earth. It is also the case that other life-forms might emerge and use their technological capabilities to create simulations of us. Once an alien civilization reaches the point of being able to create simulations, it can create lots of simulations - really unreasonably large numbers of simulated beings can be created in a universe roughly the size of ours1,2, Bostrom's estimate would be something like 10^50. And in other mathematically possible universes with the ability to do an infinite amount of computation in a finite time, you could be simulated an infinite number of times in just one universe.
One (incorrect) way of interpreting it is to think of a bunch of "worlds" spread out over the multiverse, most of them uninhabited, some containing weird green aliens, and one containing you, and saying: " Aha! I only care about this one, the others are causally disconnected from it!".
No, this view of reality claims that your current observer-moment is repeated infinitely many times, and looking forward in time, all possible continuations of (you,now) occur, and furthermore there is no fact of the matter about which one you will experience, because the quantum MW aspect of the multiverse has already demolished our intuitions about anticipated subjective experience4. Think that chocolate bar will taste nice when you bite into it? Well, actually according to Big Worlds, infinitely many of your continutions will bite the chocolate bar and find it turns into a hamster.
I once saw wormholes explained using the sheet of paper metaphor: draw two dots on a sheet of paper, reasonably far apart, imagining the paper distance between them to be an unfathomably large spatial distance, say 10^(10^100) meters. Now fold the sheet so that the two dots touch each other: they are right on top of each other! Of course, wormholes seem fairly unlikely based upon standard physics. The metaphor here is of what is called a quotient in mathematics, in particular of a quotient in topology.
But if you combine a functionalist view of mind with big worlds cosmology, then reality becomes the quotient of the set of all possible computations, where all sub-computations that instantiate you are identified. Imagine that you have an infinite piece of paper representing the multiverse, and you draw a dot on it wherever there is a computational process that is the same as the one going on in your brain right now. Now fold the paper up so that all the dots are touching each other, and glue them at that point into one dot. That is your world.
Almost all of the histories and futures that feed into your "now" are simulations, by Bostrom's simulation argument (which is no longer shackled by the requirement that the simulations must be performed by our particular descendants - all possible descendants and aliens get to simulate us).
Future Shock level 5 is "the Copernican revolution with respect to your place in the multiverse", the point where you mentally realize that perfectly dry astrophysics implies that there is no unique "you" at the centre of your sphere of concern, analogous to the Copernican revolution that unseated earth from the centre of the solar system. It is considered to be more shocking than any of the previous future shock levels because it destroys the most basic human epistemological assumption that there is such a thing as my future, or such a thing as the consequence of my actions.
Shock Level 5 is a good candidate for Dan Dennett's universal acid: an idea so corrosive that if we let it into our minds, everything we care about will be dissolved. You can't change anything in the multiverse - every decision or consequence that you don't make will be made infinitely many times elsewhere by near-identical copies of you. Every victory will be produced, as will every possible defeat.
In "What are probabilities anyway?" Wei Dai suggests a potential solution to your SL5 worries:
For example, you could get your prior probabilities from the mathematization of occam's Razor, the complexity prior. Then the reason you don't worry that your chocolate bar will turn into a hamster is that the complexity of that hypothesis is higher than the complexity of other hypotheses, such as the chocolate bar just tasting like normal chocolate. But you're not saying that this scenario is unlikely to happen: it is certain to happen, but you just don't care about it.
Wei's UDT allows you to overcome the decision-theoretic paralysis that would otherwise follow in a Big World: you think of yourself as defining an agent program that controls all of the instantiations of you, so that your decisions do matter. But remember, in order to get decisions out of UDT in a Big World, you need that all-important measure, that is a "how-much-I-care" density on the multiverse that integrates to 1.
Personally, I think that Shock Level 5 could be seen as emotionally dangerous for a human to take seriously, so beware.
However, there may be strong instrumental reasons to take SL5 seriously if it is true (and there are strong reasons to believe that it is).
1: Anders Sandberg talks about the limits of physical systems to process information.
2: Bostrom on astronomical waste is relevant here as he is calculating the likely number of people that we could simulate in our universe, which ought to be roughly the same as the number of people that some other civilization could simulate in a similar universe.
3: Not one of the originally proposed 4 future shock levels.
4: To really nail the subjective anticipation issue requires another post.