He would only travel to a mourner's home if he had died without children: Moed Katan 27b
Rushing to bring his son to school, while wearing a non-standard head-covering, in his haste: Kiddushin 30a
After a man was killed by a lion within three parasangs of Rabbi Yehoshua ben Levi, Eliyahu the Prophet refused to appear to him for three days [Rabbi Yehoshua should have prevented it]: Makkot 11a
Declaring regarding a law, "I heard it in this room!": Zevachim 13b
Family Interaction
Advice to his children: Berachot 8a
Teaching his children: Shabbat 68b
Telling his children not to prepare Matzah with Milk/Honey for the first night of Pesach: Pesachim 36a
Rabbi Yehoshua used to say that he would not stand for his son, except that his son had married into the family of the leader of the Jews: Kiddushin 33b
After Rabbi Chiyya bar Abba saw how Rabbi Yehoshua ben Levi rushed to bring his son to school, he wouldn't eat breakfast before reviewing his son's learning with him and adding some new item: Kiddushin 30a
Citations
Set of quotations from the "people of Jerusalem": Pesachim 113a
Set of declarations on the status of certain cities as walled or non-walled: Megillah 3b-4a
Two statements on the value of Torah study: Makkot 10a
How he began his discussion of the Purim Story: Megillah 10b
Citing a legal statement which Rabbi Yehoshua ben Levi made without explanation, and commenting, "This is prophecy!" [Rashi - And Rabbi Yehoshua ben Levi wouldn't have said it without a tradition from his elders]: Eruvin 60b
Rabbi Yehoshua ben Levi mentioned there were 24 instances of a certain law, and when Rabbi Elazar asked that he enumerate them, Rabbi Yehoshua ben Levi said, "Go find them": Berachot 19a
Rabbah bar bar Chana reported a lesson in the name of Rabbi Yehoshua ben Levi, to which Rav Chisda replied, "By Gd! Even if Rabbi Yehoshua ben Levi himself would say this, I would not listen to him!": Berachot 24b