And it can be hard to define what differentiates "middle" from "beginning" in a story that may not have a firm "beginning" present in the story.
I'm thinking of Stephen Hunter's The Second Saladin
as an almost-but-not-quite example. Short version: X years ago a covert op went south, a CIA op got snagged and tortured by the KGB, the Kurds he'd been working with were massacred almost completely and the woman (also American) who'd been embedded with them got out alive but traumatized. In the present, one of the surviving Kurds from that op has gotten into the U.S. to kill the man he feels is responsible for that whole debacle.
For most of the book, most of what I just described above isn't on the page in detail. Momentary flashes here and there, characters in the present talking about past events, some internal narration, but for the most part we just start at the "shit is going to be bad" part of the story and leave the preliminary off the page. Later we get a couple more in-depth flashbacks - the agent's torture in some detail, the woman's last days with the Kurds via reading her diary - so it more or less gets that on paper eventually, but in a story like this it's a little hard for me to decide if it's truly starting in the middle or if it starts at the beginning and the rest is purely backstory. It's so integral to the "now" that if it were missing the book would make almost no sense at all, so it feels
like part of the core story, but structurally...not so sure.