May one repeat words in Davening?
A. General law:
Certain words and sentences of reference to Hashem are forbidden to be repeated in close proximity if it can be viewed that the repetition is for the sake of a second deity, or if it can appear as if he is poking fun at the prayer. Although in the Talmud and Shulchan Aruch only specific phrases of repetition are discussed, nevertheless it is understood from the Poskim that one must beware from doing so by any phrase that can be viewed with such connotation. This applies even if one is not in the midst of praying.
During Davening: During Davening, it is improper to repeat any word, even if it does not have connotation of a second deity. The reason for this is because it constitutes an interval if one is found in an area of Davening in which an interval is forbidden. Furthermore, even if one is found in an area of Davening in which an interval is permitted, it is improper to repeat words of Davening, as the dialect of the prayer was given an exact number of words and letters.
Repetition due to lack of Kavana: The above applies even if one desires to repeat words due to lack of Kavana, and in such a case he is to think the words in his mind rather than verbalize them. Some Poskim however are lenient in such a case. The only exception is regarding repeating the verse of Shema/Baruch Sheim and Poseiach Es Yadeecha if one did not concentrate upon reciting it, in which case the verse must be repeated. This matter will be discussed further in a future Halacha IY”H!
It is initially forbidden to repeat Shema, whether one repeats word after word or sentence after sentence. It is even forbidden to read the entire paragraph and then repeat it a second time. It is however permitted to repeat Shema if a long interval takes place in-between the two recitations.
Repeating the first verse of Shema if one did not concentrate: If one did not have concentration when he recited the verse of Shema then he must repeat the verse quietly if he is with a congregation, in order so it not appear as if he is accepting two deities. [If, however, he is in private, then he may repeat the verse aloud.]
Repeating Baruch Sheim: If one said the verse without concentration, then he must repeat it, just as is the law regarding the verse of Shema which was said without concentration.
One who repeats the word Modim, [i.e. he says Modim, Modim] is to be hushed, just as is the law regarding repeating Shema.
May the Chazan repeat words when singing a section of the Davening, as is common by Hallel and Chazaras Hashatz of Shabbos and Yom Tov?
No. The congregation is also to refrain from doing so when singing the parts together with the Chazan.
May one repeat Modeh Ani?
Seemingly one is not to repeat Modeh Ani unless an interval has occurred in-between.
 Admur 61/8; Michaber 61/9; Brachos 33b
 So is understood from the fact the Gemara only mentions Modim and Shema, while Tosafus and the Poskim discuss the repetition of also other phrases, and whether it is allowed; So also rules Aruch Hashulchan 61/14 based on Rambam Shema 2/14 that it applies to all words and sentences. See also Igros Moshe 2/22
 Rebbe in Toras Menachem 35/60; Ben Porat Yosef p. 100, Toldos in name of Baal Shem Tov; Makor Chaim 61/11 “I have seen people repeat words 3-4 times and this is strange and very bad”; See Maharahm Shick 31; Piskeiy Teshuvos 51/6
 Tur 113 “My brother Rebbe Yechiel taught that Chassidei Ashkenaz would count and weigh the exact number of words in prayer and blessings.” See also Bach 51 that we repeat words in Az Yashir in order to have an exact number of Divine names in the prayer.
 Baal Shem Tov ibid; Piskeiy Teshuvos ibid
 See Ashel Avraham Butchach; Igara Deparka 62; Shulchan Hatahor 61/3-4; Piskeiy Teshuvos ibid footnote 56
 Admur 61/8; Michaber 61/9; Brachos 33b
 The reason: As it appears as if he is poking fun at the prayer, and in certain instances it appears as if he is accepting upon himself two deities. [see footnotes below]
 Admur ibid; M”A 61/7; Thus those that repeat Shema in Selichos each time that they say Vayaavor, may do so without worry, as there is a long interval in-between.
The reason: As when there is a long interval in-between it does not appear as if one is accepting upon himself two deities. [ibid]
 Admur 63/5; Taz 63/3; M”A 63/6; Sefer Chassidim 18; P”M 61 M”Z 3; M”B 61/22; Kaf Hachaim 61/34; See Kaf Hachaim 63/17
 P”M 61 M”Z 3; M”B 61/22; Kaf Hachaim 61/34
 Admur 61/12; Taz 61/4
 Admur 63/5; M”A 61/11 and 63/5; Levush 63/4; Ketzos Hashulchan 19/15
 Admur 121/2; Michaber 121/2; Mishneh Brachos 33b
 Rebbe ibid
 Based on Aruch Hashulchan 61/14 and Rambam Shema 2/14 that the prohibition applies to all words and sentences.