I recall the Gemara discussing which ones to double, but I'm wondering what the purpose of this is. (Sources please)
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.