During the Aseres Yimei Teshuvah, when Kaddish is recited we say "לעילא [ו]לעילא" instead of the usual "לעילא."
Why is this so? What's the reason for this practice and where did it originate?
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Levush Orach Chayyim 582:8 writes as follows:
ונ"ל שיש לו לכפול לעילא כמו שכופלין אותו בכל שאר תפילות של ר"ה וי"כ והטעם שכופלין אותו מפני שבימים הללו הוא מתעלה אחר כפול ומכופל מפי ברואיו מפני שהוא יושב על כסא דין והכל צריכין לשבחו ולפארו ולבקש רחמים על דינו
It seems to me that [the chazzan] should double le-ela, as we double it at all other prayers of Rosh HaShanah and Yom Kippur. The reason we double it is because during these days He is doubly exalted by his creations, because He sits upon a throne of judgement, and everyone needs to praise and extol him and seek mercy for his judgement.
Interestingly, Levush seems to talk about doubling le-ela on Rosh HaShanah and Yom Kippur only, and not the other days of aseret yemei teshuvah.