The Vidui that is said during the days of Selichos, and on Yom Kippur, and by Nusach Sefard on days that Tachanun is recited, starts with Ashamnu and goes letter by letter through the Aleph Bais. At the end it includes three words that begin with a Taf. Why was this set up this way, when no other letter is repeated?
Also, in the sefer Ohel Moed (late 1300s, Shaar Reishis Chochmah Derech 8), it contains only תעינו and תעתענו, and includes a second ל word, לוצצנו.
See here for a number of other sources containing this variant, including Machzorim and Siddurim from various communities.
Other added variants include מרינו, נאפנו, and many others (Google searches will again show a number of versions with those words).
Additionally, the word תעתענו is preceded by a ו in many versions, making it not countable for the ת grouping.
(These points are important because some Piyutim include doubled letters or missed letters, see comment thread above, specifically DoubleAA's comments.)
However, even if we accept that there are 3 final ת words and no duplicates throughout, הרב יאיר וסרטל answers this by explaining that תעבנו is the last sin that we mention in specific, and once we finish the Aleph-Beis, we add two general terms to the Vidui (as a sort of summary, if I understand correctly), and those are תעינו תעתענו. Thus, they are not really part of the rest of the Vidui, but we continue to use the last letter for the summary, and that is why there are 3 terms at the end.