Pirkei Avot 4: Three Crowns and One More

(יג) רַבִּי יְהוּדָה אוֹמֵר, הֱוֵי זָהִיר בַּתַּלְמוּד, שֶׁשִּׁגְגַת תַּלְמוּד עוֹלָה זָדוֹן. רַבִּי שִׁמְעוֹן אוֹמֵר, שְׁלשָׁה כְתָרִים הֵם, כֶּתֶר תּוֹרָה וְכֶתֶר כְּהֻנָּה וְכֶתֶר מַלְכוּת, וְכֶתֶר שֵׁם טוֹב עוֹלֶה עַל גַּבֵּיהֶן:

Pirkei Avot 4:13

(13) Rabbi Yehuda says: Be careful in study, for an error in study is considered an intentional transgression. Rabbi Shimon says: There are three crowns: the crown of Torah, the crown of priesthood and the crown of monarchy - but the crown of a good name outweighs them all.

The Rebbe's preliminary questions:
1. Why does it say there are three crowns, yet the crown of a good name is not included in the three? Shouldn't there be four crowns?

2. A good name is something external, what other people see about you and call you. How can it supersede and be greater than the Crowns of Torah, Kehuna and Kingship, which are all lofty spiritual, internal qualities?

3. Why does it use the world Olah al Gabeyin, ascends upon them) but means outweighs them all - and not just say "greater than?"

From the Tzemach Tzedek's explanations on Likutei Torah Shemini Atzeret:

And one can say that from Rosh Hashanah, until Shemini Atzeret, there are three crowns: On Rosh Hashanah, we "say words of kingship and this draws down the Crown of Kingship. Yom Kippur is associated with the Crown of Priesthood because the High Priest enters "the innermost part of the inner part" ie the Holy of Holies, and through his service, he causes that "Everything will bow before you". He also raises the souls towards Hashem - like it says, "When you should raise up the candles towards the center". And then, the third crown of Torah is associated with Shemini Atzeres, because it is the flow from above to below (i.e. the Torah is given to us from Hashem). And "Great is the study of Torah that brings to action" - ie the performance of Mitzvos, is the concept of the Crown of the Good Name.

שבת ל״ג ב:ו׳-ח׳

ומגליא לן אזלו טשו במערתא איתרחיש ניסא איברי להו חרובא ועינא דמיא והוו משלחי מנייהו והוו יתבי עד צוארייהו בחלא כולי יומא גרסי בעידן צלויי לבשו מיכסו ומצלו והדר משלחי מנייהו כי היכי דלא ליבלו איתבו תריסר שני במערתא אתא אליהו וקם אפיתחא דמערתא אמר מאן לודעיה לבר יוחי דמית קיסר ובטיל גזירתיה נפקו חזו אינשי דקא כרבי וזרעי אמר מניחין חיי עולם ועוסקין בחיי שעה כל מקום שנותנין עיניהן מיד נשרף יצתה בת קול ואמרה להם להחריב עולמי יצאתם חיזרו למערתכם הדור אזול איתיבו תריסר ירחי שתא אמרי משפט רשעים בגיהנם י"ב חדש יצתה בת קול ואמרה צאו ממערתכם נפקו כל היכא דהוה מחי ר' אלעזר הוה מסי ר"ש אמר לו בני די לעולם אני ואתה בהדי פניא דמעלי שבתא חזו ההוא סבא דהוה נקיט תרי מדאני אסא ורהיט בין השמשות אמרו ליה הני למה לך אמר להו לכבוד שבת ותיסגי לך בחד חד כנגד (שמות כ, ז) זכור וחד כנגד (דברים ה, יא) שמור א"ל לבריה חזי כמה חביבין מצות על ישראל יתיב דעתייהו

Shabbat 33b:6-8

They went and they hid in a cave. A miracle occurred and a carob tree was created for them as well as a spring of water. They would remove their clothes and sit covered in sand up to their necks. They would study Torah all day in that manner. At the time of prayer, they would dress, cover themselves, and pray, and they would again remove their clothes afterward so that they would not become tattered. They sat in the cave for twelve years. Elijah the Prophet came and stood at the entrance to the cave and said: Who will inform bar Yoḥai that the emperor died and his decree has been abrogated?

They emerged from the cave, and saw people who were plowing and sowing. Rabbi Shimon bar Yoḥai said: These people abandon eternal life of Torah study and engage in temporal life for their own sustenance. The Gemara relates that every place that Rabbi Shimon and his son Rabbi Elazar directed their eyes was immediately burned. A Divine Voice emerged and said to them: Did you emerge from the cave in order to destroy My world? Return to your cave.

They again went and sat there for twelve months. They said: The judgment of the wicked in Gehenna lasts for twelve months. Surely their sin was atoned in that time. A Divine Voice emerged and said to them: Emerge from your cave. They emerged.

Everywhere that Rabbi Elazar would strike, Rabbi Shimon would heal. Rabbi Shimon said to Rabbi Elazar: My son, you and I suffice for the entire world, as the two of us are engaged in the proper study of Torah. As the sun was setting on Shabbat eve, they saw an elderly man who was holding two bundles of myrtle branches and running at twilight. They said to him: Why do you have these? He said to them: In honor of Shabbat. They said to him: And let one suffice. He answered them: One is corresponding to: “Remember the Shabbat day, to keep it holy” (Exodus 20:8), and one is corresponding to: “Observe the Shabbat day, to keep it holy” (Deuteronomy 5:12). Rabbi Shimon said to his son: See how beloved the mitzvot are to Israel. Their minds were put at ease and they were no longer as upset that people were not engaged in Torah study.

The Rebbe, Shabbat Parshat Emor 1990
When one has carried out the services of Torah, priesthood, and kingship, then sharing one’s qualities with others brings about a new crown which is higher than the other ones.