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I think in your example $E[X_n]$ should be around $n^{2/3}$, see Theorem 3.12.4 of www.ime.unicamp.br/~popov/book_lyapunov.pdf; in general, this type of question can be investigated with Lyapunov functions. See e.g. Section 2.8, 3.9, 3.10, 3.12 of that book.
Essentially, Chapter 5 covers both discrete and continuous cases. For example, Theorem 5.3.1 is formulated in the continuous case, but (as noted there in the text) the results in the discrete case are the same. Up to Section 5.2, the results are general (the chain lives on a set $\Sigma$, which can be $\mathbb{R}$ or $\mathbb{Z}$).
Apart from trivial examples (say, first toss a coin, if heads - then take the above sequence, if tails - take a sequence of independent r.v. for which the SLLN holds), I don't know what to answer.
In particular, will it be enough to have a result like "w.h.p. the number of trials should be around $N^a\ln N$"? This should be easy by the usual first moment/second moment technique.
