Rav Ovadiah Yoseif (Yechaveh Da’at 2:5) states that one may never repeat words. As proof, he adduces the view of the Maharam Schick that one should never repeat words, throwing in the concept of hefsek in addition to that of possible heresy with certain repetitions, as in “modim modim”. Rav Yoseif cites Rav Feinstein among his proofs.
Rabbi J. David Bleich (Contemporary Halachic Problems, volume 2) brings down a letter published by Rav Yoseif Dov Soloveitchik in 1965 in the Cantorial Council of America Bulletin, in which he opposes the repeating of words at any point in the prayer services. Rabbi Bleich surmises that this view is based on the statement in Ta’anit 27b and Megillah 22a that we may not divide any verse that Moshe himself did not already divide (“kol pasuk d’lo pasak Moshe anan nami lo paskinan. Since the repetition of a word constitutes the creation of a new verse, it would fall under this restriction and thus be forbidden.
As a final note, there are places in the prayers where words are repeated, most famously at the end of Psalm 150 in the daily services and at the end of Psalm 118 in the recitation of Hallel. Each of these cases is dealt with by the gemara and poskim and has a special exemption from whichever issues would otherwise render such a practice forbidden (especially since both occur between two blessings, and thus hefsek might be a legitimate issue).