The Bet Yosef cites the Shiboleh Ha'leket (Rav Sidkiya Ha'rofeh, Italy, 13th century) as establishing that one should hear at least seven Kaddish recitations each day. This is inferred from the verse in Tehillim (119:164), "Sheba Ba'yom Hilalticha" – "I have praised You seven times each day."
By contrast, the Arizal maintained that one should hear at least twelve daily Kaddish recitations.
Our customary prayer service is arranged in such a way that a total of thirteen Kaddishim are recited. In the morning, "Kaddish Al Yisrael" is recited before Hodu, "Hasi Kaddish" is recited after Yishtabah, another "Hasi Kaddish" is recited after the Hazzan's repetition of the Amida, "Kaddish Titkabal" is recited after "U'ba Le'sion," another Kaddish is recited after the daily Shir Shel Yom, and then "Kaddish Al Yisrael" is recited before Alenu, for a total of six Kaddishim. At Minha, another three Kaddishim are recited – the "Hasi Kaddish" before the Amida, the "Kaddish Titkabal" following the repetition of the Amida, and another Kaddish after La'menase'ah Bi'nginot, before Alenu. An additional four Kaddishim are recited at Arbit, bringing the total to thirteen: before Barechu, before the Amida, after the Amida, and before Alenu. These thirteen Kaddishim correspond to the thirteen attributes of divine mercy. In some communities, Kaddish is not recited after La'menase'ah Bi'nginot at Minha, such that they recite a total of twelve Kaddishim, following the teaching of the Arizal.
In Ashkenazic communities, Kaddish is recited also after Alenu. This custom is followed in some Moroccan and Tunisian communities, as well.
The Ben Ish Hai (Rav Yosef Haim of Baghdad, 1833-1909) cites the Arizal as explaining how the various Kaddish recitations serve to facilitate our transition between the different spiritual realms. We cannot proceed immediately from our current realm – the realm of Asiya – to the highest realm, the realm of Asilut, where we stand before G-d and pray the Amida. We need to ascend incrementally, and it is through the Kaddish recitation that we advance from one realm to the next. The first "jump" occurs with the "Kaddish Al Yisrael" before Hodu, which elevates us to the realm of Yesira. The Kaddish after Yishtabah then lifts us to the realm of Beri'a. As no interruption is permitted during the section of "Yoser Or" until after the Amida, we ascend to the highest realm, Asilut, for the Amida prayer without a Kaddish. We then "descend" back to the realm of Beri'a with the "Hasi Kaddish" recited after the Amida, and then to Yesira with the Kaddish after U'ba Le'sion. Finally, the Kaddish following the Shir Shel Yom brings us back down to the realm of Asiya.