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For example,

Given set X = {{1,2}, {2,3}}

Would Set X be technically allowed as a subset of another set, Set Y?

Set Y: { {{1,2},{2,3}} , 8, 9}

My question in general is if this pattern can continue, or if there is a limit to "sets within sets".

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    Here $X$ is an _element_ of $Y$, not a subset. And yes, you can repeat this process as often as you like. – 2017-01-13
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    For it to be a subset of $Y $, just remove one pair of enclosing $\{\} $ – 2017-01-13

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There is no restriction in set theory of having sets as elements of other sets, as long as you avoid "the set of all sets" which leads to contradictions (Russel's paradox)

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    You also can't have a set that contains itself with the well formed theorem, – 2017-01-13
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    @QthePlatypus I don't know what the "well formed theorem" is, but you *can* have sets which contain themselves - as long as you drop the *axiom of foundation*. – 2017-01-13
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    @NoahSchweber yes I meant the axiom of foundation. – 2017-01-13
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Yes you can. Actually the set of natural numbers can be constructed this way. Refer to the axiom of infinity in ZF.