10 TIMES VIDUEI ON YOM KIPPUR

1) The levush says that the reason for viduei is because it is a day of forgiveness and atonement*. We find by every korbon that brought kapparah one would confess their sins (bamidbar 5:7) just as the kohen gadol would do on yom kippur as it says (bamidbar 16:17) vchiper bado uvad beiso uvead kol khal yisrael. In his service the kohen gadol would mention the ineffable name 10 times 3 times by every viduei (for himself for his brothers the kohanim and for the Jewish people) and once by the goralos when he would say lakshem chatas. (One could add the thought that today each one of us is the Kohen gadol doing the service in the temple)
*The Rambam says it is because it is part of teshuvah

2) The Mahril says The kohen gadol would sanctify himself by washing his hands and feet ten times throughout Yom kippur and we should intend to correspond to that as we say our viduei confession 10 times. (One could add that viduei is about purifying ourselves removing the negativity etc. so it is the same basic action of purification as washing the hands and feet.)

3) The confessions of Yom Kippur are ten on the tenth day of the month for 10 is the holy of holies (elyah rabbah)(One can add aside for the fact that ten is sheleimos etc. in the mishkan the kodesh kdashim was a cube of ten amos[in the mikdsh 20 amos])

4)When Moshe asks for forgiveness for the Jewish people after the sin of the golden calf he says remember avraham (shemos 32:13) and Rashi says Remember Abraham If [You argue that] they have transgressed the Ten Commandments, [let me remind You that] their [fore]father Abraham was tested with ten tests and has not yet received his reward. Give this [reward] to him [Abraham] so that the ten will cancel out the ten. -[from Midrash Tanchuma 24, Exod. Rabbah 44:4] For this reason we say the confession 10 times (One could add that The next pasuk says The Lord [then] reconsidered the evil He had said He would do to His people.) (kol bo)