I recall the Gemara discussing which ones to double, but I'm wondering what the purpose of this is. (Sources please)
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The Gemara you're thinking of is Sukkah 39a, which records that Rebbi used to repeat some verses of Hallel (Rashi: from אנא onwards), while R. Elazar ben Parta repeated from an earlier point - which the Gemara identifies as אודך, as is the present-day practice.
Rashi (ibid. 38a, ד"ה לכפול) says that the reason for this is that the first twenty verses in that chapter have repeated phrases (like יאמר נא in vv. 2-4, טוב לחסות in vv. 8-9) or words (like דחה דחיתני, v. 13), while beginning from אודך there's no (or less) such repetition; so we double them for consistency.
Rashbam (Pesachim 119b, ד"ה מאודך) offers a different reason. The Gemara (ibid. 119a) states that these nine verses were said by the various personages present when Shmuel anointed David as king (I Sam. 16): David said אודך, Yishai said אבן מאסו, etc. In their honor, then, we repeat each of these verses.