ANYTHING NICE: Baldur's Gate (Bioware) and Oblivion (Bethesda)
You begin Baldur's Gate by running errands (no, wait, it was the 90s, it was fun then) around the monastery where you grow up. Someone's been murdered in the stables. Someone else tries to murder you. Your mentor indicates you have a Secret Destiny, and you both slip out of the monastery at midnight - where you fall into an ambush that kills your mentor, and you flee cross-country. It'll take like a half-hour.
Towards the beginning of Oblivion, you have to escort the illegitimate imperial heir cross-country, with only one loyal retainer to help you, to a fortress-monastery on a mountaintop where the Emperor's elite guards will keep him safe, and where you can return between missions. It'll take like a half-hour.
I loved both of these openings because of, not despite, the hip-deep immersion in fantasy cliché. A hundred games start with these tropes, but very few of them have you enact these tropes, and use your navigation of the landscape and environment to settle you into the narrative.
In the interests of full disclosure, I should add that my escort duties in Oblivion required me to halt and dismount every half-mile because the heir had got hung up on a tree or something. But it didn't stop me having fun.
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'Anything Nice' is is a series of very short posts highlighting things I liked about the treatment of narrative in other people's games. It's a deliberately magpie, grab-bag approach. Some of these games I adore, some I don't even like, but I learnt from all of them.