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The Doob-Martin compactification is a canonical method of completing the state space of a transient Markov chain into a compact metric space. Characterizing the Doob-Martin compactification of a Markov chain allows us to understand the distribution of the chain conditioned at infinity and produces a characterization of all nonnegative harmonic functions on the chain. In this talk, we will discuss a characterization of the Doob-Martin boundary for Markov chains of growing ballot sequences. If time permits, we will discuss an application to the study of graph limits. Based on joint work with Hye Soo Choi, Steve Evans, and Anton Wakolbinger.