Oh man, to be honest, I have tons. Being asked to recall them specifically, though, makes it hard to sort through them in my head and pick only a couple.
We'll start with Final Fantasy Tactics. There were a lot of little optional side-quests and optional characters that you could have easily avoided entirely and still beaten the game, but I feel like a lot of depth and joy is lost by skipping them. They don't advance the main plot at all - if you take them out, it doesn't affect a thing - but having certain characters in my end-game party (or even just occasionally switching it up in random encounters with them) added an element of fun.
I think the fun was in having more choices, and in the unique abilities and stories many of these characters have. It made you WANT to take the time to make sure you could get them on your team. The stories weren't all that deep, but they were awesome tidbits in a world with an already rich story and environment.
The Mass Effect games are another example of questlines that are impactful. You can zip through the game without worrying too much about your crew's approval or relations to you, but taking the time to do each of their individual questlines just made the game that much more compelling for me. All of the stories were specifically crafted to allow you to gain insight into who each character is, what their goals and wants are, what their past was, etc, and that made the urgency of completing the mission and beating the game more real for me.
In general, as I'm sure you've by now guessed, for me personally it's all about the story. I guess that's the creative writer in me. I will go the extra mile to spend time on something if it's going to enhance the lore of whatever game I'm immersed in, if it's going to give me another character to care about or a reason to empathize (or hate!) a character I already have.
I think lots of other people who aren't interested in story probably like quests that give them something aesthetic, or something that will challenge them (like a cutscene that's hidden behind a forest filled with REALLY dangerous enemies in an old game called Lunar). For me, though, if it doesn't somehow enhance the story, I'm not usually inclined to put in the extra effort.