Why do we care about the maximally extended versions of spacetimes? One can take a spacetime and maximally extend it, so that geodesics end only on singularities, where they have to end -- not on coordinate singularities, which are not physically significant. But when we do this, we seem to get spacetimes that are not physically significant. For example, the maximally extended Schwarzschild spacetime includes all kinds of crazy stuff, including a white hole and a second copy of Minkowski space that is not actually observable even if you do happen to find yourself living inside this spacetime.
So are there strong physical or practical motivations for forming the maximally extended versions of spacetimes? Do they help us to understand things about the actual, physical universe that would otherwise be hard to understand, or are they just pure math?
I've given my own possible answer below, but I don't know if it's optimal or even correct. Other answers would be welcome.
 A: This answer to my own question is the best example of which I know that seems to show that the maximally extended Schwarzschild spacetime is of some interest other than as a purely mathematical object. This is based on a physics.SE answer by Ted Bunn, which was a summary of a longer article he and Matthew McIrvin wrote. Theirs was about a Reissner-Nordstrøm spacetime and dealt with electric charge, but I've modified it (hopefully in a correct way) to relate to the less esoteric and more physically realistic Schwarzschild spacetime. I would be interested if other people could provide better motivation for the notion of maximal extension than what I've given here. 
People often ask how the gravitational field of a black hole can escape out past the event horizon, or, in a more sophisticated formulation, how the information about its mass can get out. For a black hole that forms by physical collapse, the answer is that there is information in an external observer's past light cone about how much mass fell in -- this information comes from the mass that went into the gravitational collapse.
But the Schwarzschild spacetime is a lot simpler than one formed by gravitational collapse, and we would like to have a valid answer for it as well. The answer here is that we form the maximal extension, which includes a white hole singularity. The information about the mass is on the observer's past light cone, which contains this singularity.
A: I think there is a misconception regarding the role of extending spacetimes. People often consider maximal extensions not to avoid coordinate singularities but rather non-physical spacetimes such as the one you get by starting with Minkowski spacetime and removing a single point. If you take a point to the past of the removed one there appears funny causal curves because there is a future directed geodesic which ends at a finite time not because there is a singularity, but becase you artificially removed that point. Since there is no guarantee from Einstein Equations that starting with smooth data these kind of behavior will not occur it is usual to consider maximal extensions. Notice that in this example of Minkowski without a point the "singularity" would not be a coordinate one, just not very physical. You can find a complete discussion of these things in Wald's GR book, chapter 8.
Now, from the point of view of the physical spherical collapse you will end in fact with Schwarzschild metric (asymptotically), but considering the full spacetime you will never see the white hole or the other asymptotically flat end that you mentioned, so it is true that the maximal extension might not be physical (both points you made in the answer you proposed if I understood correctly). But there is this paper from Wald and Racz (http://arxiv.org/abs/gr-qc/9507055) that says that under some conditions (very reasonable ones, although there is a caveat for extremal black holes) if you have a realistic black hole you can always extend the horizon to obtain something like the other parts of Schwarzschild spacetime, and the physics will be same in the sense that what you get from calculating in the extension (in the real flat end and inside the black hole) will yield the true results. So in a sense the maximal extensions are not physical, but studying them will lead to to physical results, and maybe it's easier to evaluate some things in this way. Not sure if this is satisfactory as an answer though.
EDIT: The previous discussions is not precise in the sense that the gravitational collapse is fully dynamical and the metric is never stationary (the spacetime does not possess timelike Killing vectors). But numerical results suggest that if you start with a dynamical collapse (not necessarily axi-symmetrical) then after the black hole forms the metric will be, asymptotically to the future, represented by the Kerr metric, or in other words that perturbations will decay exponentially. In this sense the late stages of realistic gravitational collapse should be described by a stationary black hole. As already noted this black hole will not have the white hole or the other flat end, but the result by Racz and Wald shows that if you have a stationary solutions then you can always form a bifurcated horizon (which corresponds to a spacetime with this other non-physical regions). Thus any result which depends on bifurcated horizons are meaninful for the late stages of gravitational collapse. Trimok's comment is one good example of such results. A good paper discussing the role of bifurcate horizons for the appearance of thermal states is this one by Kay and Wald (http://www.sciencedirect.com/science/article/pii/037015739190015E). So the idea is that various semiclassical (and some classical as well) features of black hole dynamics are better described by the maximal extension, and if you consider that at late times the black hole should be well described by a stationary one then you can always study the maximal extensions, which provide different insights.
