The minhag is that we remove the top ring from the lulav on Hoshanah Rabbah (see Shulchan Aruch 664:1)
The sefer Tiferes Naftoli cites a moshul which gives some insight into this custom: There was once a king who had a naughty son who wasn’t behaving the way a prince should behave. The king sent him to a far- away town, in order that the prince be upset that he’s been sent away from his father the king, and hopefully through this he will start to behave the way a prince should. When the prince arrived in the far away town, they weren’t very nice to him. They didn’t do anything good for him, they hurt him, starved him and even took off his royal clothes and dressed him in rags.
The king was waiting eagerly to hear how his son had changed and hopefully would want to return home. A year went by, and the king couldn’t wait any longer, so he went to visit his son to see what he was up to and how he was being treated. When the king came to town, everyone realized that the king had come, and they quickly removed the rags off the prince, washed him, dressed him with royal clothes, and gave him what to eat, in order that he would look well-kept in the eyes of the king. When the king saw how they were treating him so nicely, he said to himself, “it’s best if I leave him here a bit longer, it will definitely turn him into a better person, especially as they are treating him so nicely”.
The king sent a message to his son, that tomorrow he will be going home, and before he leaves, he would just like to say goodbye to his son. When the son heard this, he became very upset, he was upset that the king was going to leave him behind and let them continue bullying him.
The prince went to ask advice from his wise friend. The wise man told him, the reason your father is leaving you here, is because he thinks you aren’t lacking anything, he thinks you can act like a king here. If only he would know how badly they are treating you, he would take you back home with him, you must find a way to show you father how badly they really treat you.
The next day the prince took off the royal garments, and put on the rags he was forced to wear the whole time, and bent down and pleaded to his father the king. “Please see how they treat me the whole year, please take me home, they only dressed me nicely because you were here, the whole year they are mean to me, please take me home”.
The nimshal is, Hakodosh Boruch Hu sent us to the land of our enemies, hoping this would make us do teshuvah. Hashem sent us into golus [exile] hoping we would do teshuvah, unfortunately we don’t seem to be getting the message, we accustom ourselves to live like them, and act like them etc. When it comes to yemai rachamim and ratzon, Hashem comes and checks upon us to see how we are behaving. During the yemai rachamim and ratzon, we come to shul, we daven and do lots of mitzvos, and lots of good deeds. Just like the lulav only has a lev echod – one heart, similarly, we have only one heart towards Avinu Shebashomayim in these days, and Hakodosh Baruch Hu is very happy with us.
On Hoshanah Rabbah, Hashem departs from us, as now the days of rachamim and ratzon are over, Hakodosh Baruch Hu wants to leave us in golus, amongst the goyim. Therefore, we remove the rings from our lulavim, to be meramaz [to hint] to Hashem, just because now we do lots of Torah and mitzvos, we can’t keep it up the whole year, the goyim cause us so much trouble. Just like this lulav is now without any rings – the lulav is untied, similarly us, just because now we look good, but it’s only now, the rest of the year it’s very difficult for us. Please don’t leave us here, we need you, we can’t live without you, Hoshanah – please save us.
