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:
All possible worlds are real, and probabilities represent how much I care about each world. (To make sense of this, recall that these probabilities are ultimately multiplied with utilities to form expected utilities in standard decision theories.)
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.