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Umineko no Naku Koro ni: For Ep 1-4 the question is about who the killer is. Is it one of the family members or servants, is it a 19th person or is it the Golden Witch Beatrice? For Ep 5-8 the question changes into Who is Beatrice?
Naruto: Who the hell is Tobi?. Alternativly, How can True Peace be reached?
One Piece: What is "One Piece"? Is it treasure? Or is it something Gold Roger personally treasured?
For Robin it is, "What happened during the void century?" and for Ace, "Did I deserve to be born?"
The Scribbler: Is Suki, who suffers from Split Personality disorder, killing patients at her high-rise halfway house and making them look like suicides? Or is it — as she insists, one of her personalities, the titular Scribbler?
Each Harry Potter book has its own.note As a bonus, some of these are trick questions.
Philosopher's Stone: What is Snape after?
Chamber of Secrets: Who opened the Chamber?
Prisoner of Azkaban: What did Black do, precisely?
Goblet of Fire: Who put Harry's name in the Goblet of Fire?
The 1st of the Otherland books uses several of these at once. What is happening with Rene's brother? Why is Paul switching dimensions? What is the purpose of this Grail organization, and who is their leader? What Mr. Sellars is up to, and many others. Because this book is widely considered to be an oversized prologue to the later ones, the answers are pretty well known.
The Chronicles of Amber (Series One): Who is Really Behind It All (where what "It All" is varies over the course of the series; a few are "Corwin winding up in the hospital", "what happened to Oberon", "the death of [insert name here]" (multiple times), "the attacks on Amber"). Some of these actually get answered, others turn out to be red herrings, irrelevant, or to have had multiple causes.
The Isaac Asimov story "The Last Question": Can entropy be reversed?
In Charlie and the Chocolate Factory one reason the Golden Ticket contest is such Serious Business is because the finders will learn the answer to the question "How does Mr. Willy Wonka, who has been a recluse for ten years, manage to keep his factory producing the best candies in the world when no workforce is ever seen entering or exiting it?"
Season 3. What's the deal with Miguel/Who is the 'Skinner'?
LOST thrives on this. The Myth Arc is about the protagonists getting dropped in the middle of these strange, inexplicable and in some cases scientifically impossible occurrences and trying to puzzle out the answers to these questions. And there are many questions: "Where are we?" "What is the monster?" "How are people being healed?" "Will we ever escape the Island?" "What is the Island?" "Who are The Others?" "What's in the hatch?" "Should they keep pushing the button?" "What crashed the plane?" "What are the Whispers?" And due to the focused viewpoint, a lot of them don't get explicit answers. Season 6, however, cuts to the heart of it and makes it clear that the show's Driving Question is actually "Why are these people on the Island?" The answer has many facets that touch on both the A-story of the show and the thematic drive of the show.
Season 3: Both "Who is the serial rapist?" and "Who killed Dean O'Dell?"
In the first season of The Sarah Connor Chronicles, the Driving Question was how to find and destroy the Turk. By the second season, the question instead came in the form of a Room Full of Crazy drawn by a dying resistance soldier, and the cast's attempts to figure out the meaning of all the names and places written there.
And the overarching Terminator question: How can SkyNet be stopped?
The Prisoner: Who is Number One? Who runs The Village? Why did Number Six resign? (This is an interesting example, because whoever runs the village knows who Number One is and wants to know why Number Six resigned, Number Six knows why he resigned, but doesn't know the answers to the first two questions, and the audience knows pretty much bupkis ... and it's never explicitly cleared up either.)
Ashes to Ashes: What happened to Sam Tyler? (Whilst this was only an actual Driving Question in the third and final series, it had been brought up very briefly in the first episode of the first series.)
Who is behind and what is the true purpose of Project Centipede?
What really happened after Coulson's "death" at Loki's hands in The Avengers?
What is the truth of Skye's mysterious past, and how is S.H.I.E.L.D. connected?
The Coulson question has been addressed, but has led to another question — what's the deal with the alien corpse in the Guest House facility, and why is it being kept such a secret that you have to be on Nick Fury's clearance level to know about it?
Apocalypse World: "What caused the Apocalypse, and how can we fix it?" is recommended, though the answer of course is up to the MC and the players.
Planescape: Torment. You wake up without knowledge of who you are, what you're doing here, and why you can't remember anything. The first order of business is to fix that, and it takes most of the game.
The Arc Words "What can change the nature of a man?" would also qualify. Also a double subversion. Ravel tells you the answer doesn't matter, and it's your choice. At the very end, you find out it's Belief.
Baldur's Gate: Who is the "armoured figure"? Why is he trying to kill the Player Character? What do all these dreams and portents mean? And what is the Iron Throne trying to do and what does that have to do with any of the other questions?
In the sequel it's "Who is Jon Irenicus and what does he want from you?"
Team Fortress 2: What are the sordid secrets of the Mann Family and their inheritors? Who is the Unseen Man in the photograph? Why is the Administrator involved? What are Dell Conagher's motives? What, or who, is The Pyro?
Metal Gear has several, the most prominent being "Who are the Patriots?" and perhaps "Who is Ocelot really working for?"
Gunnerkrigg Court: What exactly is the Court? (Which in turn is connected to several sub-questions: What was Surma's and Anthony's history with the Court? Who is that ghost in the gorge, and why is she trapped there? What the hell is Zimmy's deal?)
Sluggy Freelance: What is Oasis? It doesn't drive the whole comic, but it does several stories in it, and remained unresolved for a long, long time. There are also other questions that drove single storylines but have been resolved, eg. "Where is Teknokon One?" and "What are the ghouls?" (There are also other questions, such as "What is Bun-bun's past?" that are just as mysterious but don't actually drive the story.)
What hides in the Silent World? This one drives the main plot of the comic.
El Goonish Shive: Why are the French Immortals amnesic? Who summoned the flame summon and why? The last question drives the New And Old Flames storylines. Other big questions exist but have been resolved, their solutions are not being actively pursued in universe or are not pondered by the characters themselves.
In Inverloch, the plot starts with the simple question "What happened to the elven boy who disappeared 12 years ago?". But even before Archeron can start his journey to find out, the even bigger question is "Why does everyone in his village seem to know exactly what happened, but nobody is willing to tell him before he find out himself?".
In morphE the seedlings have two gigantic ones that fuel their efforts. "Why does Amical want to train new mages?" and "What happened that caused everyone to wake up in crates with no memory of how they got there?". As they are captives in Amical's manor, the first question is the one which drives them through their mage training. The second question is what will drive them beyond there.
Homestuck progressed pretty quickly to "What the hell is going on?" and while there are elements of that persisting throughout, more recent driving questions are "who/what are the trolls", "who/what is Doc Scratch", "who/what is Lord English", and "can Lord English be defeated." There's also the lesser questions of "can Jack Noir/the Condesce be defeated", though at least we know who/what they (the Condesce is Betty Crocker, who is actually Alternia-Meenah).