Shemini Atzeres
The Midrash says that after seven days of Succos, Hakadosh Baruch Hu says, קשה עלי פרידתכם"It is hard for Me when you leave," and Hashem asks us to stay one more day. The Imrei Emes writes that the feeling should be mutual. Bnei Yisroel should also feel that it is difficult to part from the Yomim Tovim.
The Rema MiPano teaches that the literal translation of חג is a circle. All Yomim Tovim are a chag, a circle, surrounding a primary, focal point. The focal point, he said, is Shemini Atzeres. This is the reason, according to some opinions and siddurim, Shemini Atzeres isn't referred to as a חג (see Shulchan Aruch 668). All Yomim Tovim חג שברג, surrounding this special day, which stands in the middle.38
It states in Mishlei עוז והדר לבושה ותשחק ליום אחרון. The Vilna Gaon zt'l explains: עוז is Rosh Hashanah because Hashem judges us with His might. והדר is Succos, the day we take a פרי עץ הדר. And then comes Shemini Atzeres, ותשחק ליום אחרון the day of immense happiness.
The Gemara (Taanis 7) discusses גדול יום הגשמים "How great is the day it rains!" The Avodas Yisrael zt'l says that the Gemara is referring to the greatness of Shemini Atzeres, the day we daven for rain. (We say תפילת גשם).
The Chasam Sofer zy'a teaches that Shemini Atzeres is greater than Yom Kippur. Yom Kippur, we love Hashem through afflictions (fasting, etc.). Shemini Atzeres is to love Hashem through joy, which is a greater connection, as it states (Shir HaShirim 7:7( מה יפית ומה נעמת אהבה בתענוגים "How beautiful it is when the love to Hashem is with pleasure and joy."
The Chinuch (324) writes that we don't take the four minim on Shemini Atzeres because the four minim are to remind us to direct the joy we have in our hearts at this time of year (from gathering in the crops) to Hashem. On Shemini Atzeres, we don't need any reminders to direct our love to Hashem because 'יום השמיני כולו לה "The eighth day is solely for Hashem." It is a day we are entirely with Hashem, and our love and joy are fully with Him.
The Yesod Yosef writes, "The custom is to take all the sifrei Torah out of the aron kodesh... Because when a father loves his son, he opens his treasuries for him and doesn't withhold anything. Similarly, Hashem bestows upon us immense goodness."
The Zohar states, "On Hoshana Rabbah, the judgment is completed, and the blessings begin. On the following day (Shemini Atzeres), the Jewish nation celebrates with the King and receives blessings for the entire year. It is only the Jewish nation that is present at that celebration. And when one sits alone with the king, whatever he asks will be granted."
Reb Chaim Palagi zt'l teaches, "Be extremely careful with the tefillos of Shemini Atzeres to say them with immense kavanah... There is no better day and greater eis ratzon than this day, when Hashem desires to hear our tefillos.... Everything depends on this day."
The Kotzker zt'l said that on Shemini Atzeres is the gzar din (the decree) for parnassah. His son, Rebbe Dovid zt'l, used to daven b'yichidus (by himself). One year, on Shemini Atzeres, the Kotzker called for his son and told him that on Shemini Atzeres, it is important to daven with a minyan because, on this day, it is decreed how much parnassah one will receive in the coming year. This is hinted at in the words והיית אך שמח. The final letters of והיית אך שמח spell חת"ך the name of the malach appointed over parnassah. Chazal tell us that the words והיית אך שמח refer to Shemini Atzeres.
Reb Aharon Chernobyler zt'l said that benching גשם on Shemini Atzeres (said with Mussaf) is mesugal for parnassah.
The Rayatz of Lubavitz zt'l said in the name of his father, the Rebbe Reshab zt'l, "We must cherish the forty-eight hours of Shemini Atzeres and Simchas Torah. Every moment of these days, we can fill buckets with ruchniyus and gashmiyus. We receive this bounty because of the dancing. (Sefer Maamarim 5711, p.79).39
38. At a chasunah, there is a chuppah, dancing, and a meal, but the purpose of it all is for the chasan and kallah to go home and to build a home together. The Sfas Emes says that the yomim tovim represent various stages of a chasunah. For example, the schach on the succah resembles the chuppah. After the chuppah is the yichud, which is an even greater unity – that occurs on Shemini Atzeres.
39. The Rayatz zt'l of Lubavitz said that during the hakafos, we can ask for great things, and our tefillos will be answered. When we dance with the Torah, we are demonstrating our love for Hashem and His Torah, as we say in the Hoshanos,חבוקה ודבוקה בך "hugging and connected with You". On Succos, one person holds the sefer Torah, and the congregation, holding the four minim, circles the sefer Torah and the bimah. On Simchas Torah, we show a higher level of connection to Hashem because people hug the Sefer Torah and dance with it around the bimah.
When a storeowner trusts his customer to take merchandise and pay later, this is called הקפה. The Rayatz explains how this alludes to the הקפות of Simchas Torah. He explained that a storekeeper will only extend credit to customers he trusts and who are able to pay up. If he suspects a customer won't pay, he won't sell to him on credit. The dancing on Simchas Torah is called הקפות because, on Simchas Torah, Hashem gives us salvation and many brachos on account of our future good deeds. On this day, Hashem trusts us that we will pay up our debts, so he gives us on loan. [It seems that this is because Simchas Torah is after the Yomim Noraim, and every Yid was inspired to do teshuvah. We have repaired our credit, and Hashem gives us salvations and brachos, on account of our good deeds to come throughout the coming year.]