Which religions seek to actively convert people (besides Christianity & Islam)?

[DavidB.]

Judaism is basically a religion only for Jewish people. Jews acknowledge most of Jewish law doesn't apply to outsiders, and that outsiders can still find favor with God without converting (law of Noah).

This may be true (though not only can non-Jews "find favor with G-d" by keeping the Noahide laws, they are actually supposed to keep them), but let's not forget we say this in Aleinu three times a day (bolded emphasis by me):

"Therefore we put our hope in You, Hashem our G-d, that we may soon see your mighty splendor, to remove detestable idolatry from the earth, and false gods will be utterly cut off, to perfect the universe through the Almighty's sovereignty. Then all humanity will call upon Your Name, to turn all the earth's wicked toward You. All the world's inhabitants will recognize and know that to You every knee should bend, every tongue should swear. Before You, Hashem, our G-d, they will bend every knee and cast themselves down and to the glory of Your Name they will render homage, and they will all accept upon themselves the yoke of Your kingship that You may reign over them soon and eternally."

This is to say -- Judaism is not entirely like "Judaism's for Jews, and everybody else can do whatever they like." At some point in the future non-Jews are supposed to recognize G-d as the Almighty too.

[DavidB.]

Judaism is basically a religion only for Jewish people. Jews acknowledge most of Jewish law doesn't apply to outsiders, and that outsiders can still find favor with God without converting (law of Noah).

This may be true (though not only can non-Jews "find favor with G-d" by keeping the Noahide laws, they are actually supposed to keep them), but let's not forget we say this in Aleinu three times a day (bolded emphasis by me):

"Therefore we put our hope in You, Hashem our G-d, that we may soon see your mighty splendor, to remove detestable idolatry from the earth, and false gods will be utterly cut off, to perfect the universe through the Almighty's sovereignty. Then all humanity will call upon Your Name, to turn all the earth's wicked toward You. All the world's inhabitants will recognize and know that to You every knee should bend, every tongue should swear. Before You, Hashem, our G-d, they will bend every knee and cast themselves down and to the glory of Your Name they will render homage, and they will all accept upon themselves the yoke of Your kingship that You may reign over them soon and eternally."

This is to say -- Judaism is not entirely like "Judaism's for Jews, and everybody else can do whatever they like." At some point in the future non-Jews are supposed to recognize G-d as the Almighty too.

Come on, David, you know as well as I do that Judaism calls on outsiders to worship Him and shun false gods at the same time they don't want the same outsiders to convert to Judaism. The religious Jewish perspective is that they have certain obligations to God as His Chosen People and non-religious Jews aren't bound by those obligations, but that those outside people can still win favor by renouncing false gods and worshiping Him.

Yes, I almost entirely agree -- where does your statement contradict mine? I'm not saying Aleinu says everybody is supposed to convert to Judaism at some point. I'm saying Aleinu says everybody is supposed to recognize G-d as the Almighty. The part where I disagree is where you say that "non-religious Jews aren't bound by those obligations" -- perhaps you meant to say non-Jews? Because non-religious Jews are certainly supposed to keep the mizvot just as much as a Haredi person (though this is different for someone who doesn't know about his Judaism or was raised without any knowledge of it). Non-Jews are bound by the Noahide laws but obviously not by the 613 mitzvot.