Final Fantasy 6 is one of my top five favorite games of all time, and it has a lot of little details in it that you might miss if you blink. I love how each character has his or her own theme song, and even remixes of the same song depending on the mood. Even the last part of Dancing Mad, the final boss theme, is just a remix of Kefka's orignal song.
Sometimes in FF6, you can change out your party members, and depending on who you might have in your party, some characters may say different things which might add insight to the story you might've not known otherwise. My favorite is when you trick the gambler Setzer with a one sided coin. There is an earlier cutscene where Edgar and Sabin use a coin to decide who will be the king, and if Edgar and Sabin are in your party when you trick Setzer, Edgar flips the coin and Sabin says something like, "Brother, that coin! It couldn't be..."
One final thing in FF6. In the snowy mining town of Narshe, some of the townsfolk claim they've seen a Yeti. That Yeti is Umaro of course, but you can spot him, too, peeking out of holes in the walls in caves, long before he can join your party!
In other games, character standing animations are also fun details. Some of my favorites are in Mega Man X. Normally he just blinks his eyes when standing still, but if he's low on energy, the light on his helmet flashes red, and he'll breathe heavier. And if you are on Chill Penguin's stage, you can even see frost smoke come from his mouth like how you can see your breath when it's cold. Never though robots had to breathe, but this is a video game so oh well.
One last standing animation I like is in Donkey Kong Country 2. One of the stage themes is a giant beehive, and you can stick to the walls if honey is on them. While sticking to the wall, sometimes Dixie Kong will put her finger on the honey and then lick it.
OK sorry for being so long. By the way, I just posted a new podcast that I hope you all can listen to. It's about Transformers, so I think you'll like it.
https://open.spotify.com/episode/07zOFGLATchbGTpZBXuOO8