Bnei Yissas’char, though, not completely satisfied with the above answer, asks again: Why do we specifically recite the verses three times? He answers as follows: “It is written (Psalms 119:165), ‘Shalom rav le’ohavei toratecha ve’ein lamo michshol – Manifold peace to the lovers of Your Torah, and they shall not encounter any stumbling blocks.’ This is in accord with what our sages (Gittin 46a) state: yamim (days) refers to two, whereas the term rabbim (many) refers to three. Therefore the use of ribbuy [lit. rav], the plural count, refers to three. The verse actually means that when shalom is recited many times – [at least] three – no stumbling block will be encountered.”
Therefore, at the monthly renewal of the moon, we say “Shalom Aleichem” three times to ensure that we don’t encounter any stumbling blocks during the new month.
Bnei Yissas’char also points out that Tanach calls the moon by three different names: yare’ach, levana, and sahar. This, too, serves as a reason to recite “Shalom Aleichem” three times – once for each of the moon’s names.
We find yet another explanation in Bnei Yissas’char (found in this discussion as well as in Ma’amar Kislev 13). He suggests that we say “Shalom Aleichem” three times for the three fundamental mitzvot: Kiddush Hachodesh, Shabbat, and Milah. (It was these three mitzvot whose observance the Syrian Greeks attempted to squash, and thus, all Torah observance – see Megillat Antiochus.)
Interestingly, we conclude Kiddush Levanah with the prayer of “Aleinu Leshabe’ach,” the same prayer that concludes Shacharit, Minchah, and Ma’ariv. Rabbi Yisrael Chaim Friedman of Rachov (Likutei Mahariach, order of Rosh Chodesh) explains that we do so because it is well known that Joshua composed this prayer of thanks upon conquering Jericho, and our Sages (in Bava Batra 75a) compare Moses and Joshua to the sun and moon, respectively. The face of Moses – the teacher – is compared to the sun, and the face of Joshua, – his student and thus his reflection – is compared to the moon.
Rabbi Friedman also offers another explanation. Since we “danced” before the moon during Kiddush Levanah, we may have accidentally given the impression that we were – heaven forbid! – praying to the moon. We therefore conclude with “Aleinu Leshabe’ach” which clearly indicates, without leaving any doubt, that our entire prayer was to “Adon Ha’kol – the Master of all [creation].”
With the Likutei Mahariach’s explanation, we can now also understand why we conclude each of our three daily prayers with Aleinu. We want to confirm that each and every word that we prayed from the outset of that prayer was directed only to G-d.
In concluding, I would like to note that the Aleinu prayer itself contains a triple repetition: “shelo asanu k’go’yei ha’aratzot – that He has not made us like the nations of the earth,” “shelo asanu k’mishpechot ha’adamah – that He has not made us like families of the earth,” and “shelo sam chelkeinu kahem ve’goralenu k’chal hamonam – that He has not made our portion like theirs, nor our lot like their multitudes.” All three verses are basically making the same statement: that G-d has chosen us and given us a standing unique among all the nations, from our very beginning and for all eternity.
What a beautiful prayer with which to conclude a prayer that beckons our souls to cleave to Hashem, repeatedly, every single month, as we await our final deliverance with great anticipation, speedily in our days!