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There is nothing special about the symbols $\pm 1$ in the question; the spaces of binary strings $(\mathbb{Z}/2)^\mathbb{N}$ and $(\mathbb{Z}/2)^\mathbb{Z}$ are a more standard notation. As Terry says, you can switch between singly infinite and doubly infinite sequences.

The process of taking pointwise limits of translates over and over again until no more data can be removed is exactly equivalent to finding a minimal set with respect to the dynamical action of the shift map. If $x$ does not lie in a minimal set, then the closure of its orbit contains a $y$ whose orbit has a smaller closure; thus $y$ is a limit of translates of $x$ in which something has been lost. On the other hand, all points in a minimal set are limits of translates of each other, so once you are in a minimal set, there is no way to erase any more data.

Birkhoff established that a symbolic sequence $x$ is minimal, or lies in a minimal set, if and only if it is "almost periodic". Anton used the term "quasiperiodic", but this is a confusing term that sometimes means almost periodic and sometimes means other things. It is true that a quasiperiodic pattern such as a Penrose tiling or a quasicrystal is almost periodic. (These are higher-dimensional examples, but the issues are the same for all locally compact abelian groups.) Sometimes quasiperiodic examples have the additional property that the recurrence length $L(\ell)$ is linear in $\ell$ or $O(\ell)$. Penrose tilings are already related to one interesting class of examples in one dimension: If $\alpha$ is an irrational number between 0 and 1, then the sequence $a_n = \lfloor (n+1)\alpha \rfloor - \lfloor n \alpha \rfloor$ is an almost periodic binary sequence.

What Anton could mean by the statement "you will not get more" is two things. First, that every almost periodic sequence is minimal, and therefore that you can't simplify it further by taking a limit of its translates. Second, that there isn't any simpler characterization of minimal shifts than that they are almost periodic.

Section 13.7 of the book "An introduction to symbolic dynamics and coding" by Douglas Lind gives a survey of properties of minimal shifts in symbolic dynamics. Lind says that that a widely studied example is the Morse-Thue sequence, which is an example of an almost periodic sequence obtained by substitution rules. Lind says that any substitution-type almost periodic sequence, or substitution shift, has zero entropy, but that Furstenburg found an example of a minimal shift with positive entropy. This suggests that it is not always possible to "get rid of the randomness", as Tim (?) asks, even though all limits are highly structured in the sense of being almost periodic.

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There is nothing special about the symbols $\pm 1$ in the question; the spaces of binary strings $(\mathbb{Z}/2)^\mathbb{N}$ and $(\mathbb{Z}/2)^\mathbb{Z}$ are a more standard notation. As Terry says, you can switch between singly infinite and doubly infinite sequences.
The process of taking pointwise limits of translates over and over again until no more data can be removed is exactly equivalent to finding a minimal set with respect to the dynamical action of the shift map. If $x$ does not lie in a minimal set, then the closure of its orbit contains a $y$ whose orbit has a smaller closure; thus $y$ is a limit of translates of $x$ in which something has been lost. On the other hand, all points in a minimal set are limits of translates of each other, so once you are in a minimal set, there is no way to erase any more data.
Birkhoff established that a symbolic sequence $x$ is minimal, or lies in a minimal set, if and only if it is "almost periodic". Anton used the term "quasiperiodic", but this is a confusing term that sometimes means almost periodic and sometimes means other things. It is true that a quasiperiodic pattern such as a Penrose tiling or a quasicrystal is almost periodic. (These are higher-dimensional examples, but the issues are the same for all locally compact abelian groups.) Sometimes quasiperiodic examples have the additional property that the recurrence length $L(\ell)$ is linear in $\ell$ or $O(\ell)$. Penrose tilings are already related to one interesting class of examples in one dimension: If $\alpha$ is an irrational number between 0 and 1, then the sequence $a_n = \lfloor (n+1)\alpha \rfloor - \lfloor n \alpha \rfloor$ is an almost periodic binary sequence.