Am I a complete freak in that I don’t pay any special attention to the first line of a book?
Oh well… Add another way to the list.
Yes.
Yes, you are. 
But really, I am bumping this thread to say:
I didn’t like much of the book, but oh! that first line is enough to want to bump this thread.
**Whether I shall turn out to be the hero of my own life, or whether that station will be held by anybody else, these pages must show. **
This is the first sentence of David Copperfield. Why wouldn’t he be the hero of his own life? If not he then who? And then the reader is asked to make the decision. How can you not want to make that decision and read the book so as to make it?
“Thursday, March 17,I spent the morning in anxiety,the afternoon in ecstacy,and the evening unconscious.”…
Risk by Dick Francis
Since mine have already been mentioned, I submit:
“Lifetimes ago, under a banyan tree in the village of Hasnapur, an astrologer cupped his ears–his satellite dish to the stars–and foretold my widowhood and exile.” -Jasmine by Bharati Muckherjee
“The conversation drifted smoothly and pleasantly along from weather to crops, from crops to literaturs, from literature to scandal, from scandal to religion; then took a random jump, and landed on the subject of burglar alarms.”
The McWilliamses and the Burglar Alarm, a short story by Mark Twain. Nearly my favorite short story of his.
Here’s the whole text. Don’t read it while drinking anything!
Not the first line of the book, but of the Introduction, so I’m counting it. Unfortunately, it is from memory, but I think I have it fairly close:
“The reader will start this book as an act of faith, but will finish it as an act of charity”.
John Barth, Giles Goat Boy
Not a big Barth fan, but this got me to buy the book (so don’t underestimate the value of an opening sentence!)
Another vote for the Chandler quote. This is what Lou Grant read to Mary Richards when he was talking about what good writing can do.
“There was a boy called Eustace Clarence Scubb, and he almost deserved it.”
-The Voyage of the Dawn Treader, C.S.Lewis
The opening lines of a comic book are very different than the opening lines of a book, simply because the accompanying imagery can convey meaning not put to words. But here’s a smattering.
As well as giving baths for a living in those days, Chelo was also a midwive. She can tell you stories. Heartbreak Soup, from Love and Rockets by Gilbert Hernandez
I have the home stretch all to myself when the readings stop making sense. I switch to manual – “Bruce this is Carol. You’re going too fast!” – but the computer crosses it’s own circuits and refuses to let go. I coax it. The Dark Knight Returns, Frank Miller
Dog carcass in alley this morning, tire tread on burst stomach. This city is afraid of me. I have seen it’s true face. Watchmen, Alan Moore
There were voices, and thunderings and lightnings… and there followed hail and fire mingled with blood. Kingdom Come, Mark Waid
This one is worn and dull like an old coin. But patches here and there still glint. If I rub them carefully I can make them shine. Elektra, Frank Miller
It’s raining in Washington tonight. Plump, warm summer rain that covers the sidewalks with leopard spots. Downtown, elderly ladies carry their houseplants out to set them on the fire escapes, as if they were infirm relatives or boy kings. I like that. Swamp Thing, Alan Moore
“The sun did not shine, it was too wet to play. So we sat in the house all that cold, cold wet day.”
Dr Seuss, The Cat In The Hat
“If you are interested in reading books with happy endings, then you would be better off reading some other book.”
Lemony Snicket, A Series of Unfortunate Events
“Man,” said Terl, “is an endangered species.” Battlefield Earth by L. Ron Hubbard
Damn youuuuuuuu! That was so mine! 
Twas a long time ago, longer now than it seems
In a place perhaps you’ve seen in your dreams.
For the story you’re about to be told
began with the holiday worlds of auld.
Now you’ve probably wondered where holidays come from.
If you haven’t I’d say it’s time you begun.
“Hale knew, before he had been in Brighton three hours, that they meant to murder him.”
Graham Greene, Brighton Rock
The classic hook. How do you *not * keep reading with a book that starts that way? This is one of the more “gimmicky” of Graham Greene’s opening lines, but no less worthy for its melodramatic overtones, imo. (The opening paragraph of *The Confidential Agent * is subtler, more nuanced. And amazingly, no less compelling. Don’t have a copy handy or I would have cited it here.)
“I owe the discovery of Uqbar to the conjunction of a mirror and an encyclopedia.”
Jorge Luis Borges, “Tlon, Uqbar and Orbis Tertius”
Borges’ tone is always so subdued, yet suggests these fantastical vistas waiting only inches away. This story played with my brain like a kid going to town on his first ball of Silly Putty.
“Francis Marion Tarwater’s uncle had been dead for only half a day when the boy got too drunk to finish digging his grave and a Negro named Buford Munson, who had come to get a jug filled, had to finish it and drag the body from the breakfast table where it was still sitting and bury it in a decent and Christian way, with the sign of its Savior at the head of its grave and enough dirt on top to keep the dogs from digging it up. Buford had come along about noon and when he left at sundown, the boy, Tarwater, had never returned from the still.”
Flannery O’Connor, The Violent Bear it Away
This opening passage is memorable to me for two reasons, I think: one, because O’Connor throws so much at you so quickly and, seemingly, artlessly, that you’re rooted to the spot until the paragraph ends (I was, anyway). But two, as the book continues, she does a nifty thing. Throughout the following fifty pages or so, she simply retells exactly what was in that first paragraph, though obviously expounding on it in greater detail. It’s like she blurts out a huge portion of the novel to you in two sentences, and when she’s gotten your attention, she says, Now. Here’s what I meant by that. I always found that cool.
“Jack Torrance thought: Officious little prick.”
Stephen King, The Shining
There’s something to be said for brevity, too. 
“They murdered him.”
Robert Cormier, The Chocolate War
“The winter before he was sixteen, Pup sold his soul to the devil.”
Ruth Rendell (again) The Killing Doll
“El día que lo iban a matar, Santiago Nasar se levantó a las 5,30 de la mañana para esperar el buque en llegaba el obispo.”
-Gabriel Garcia Marquez’s “Cronica de Una Muerte Anunciada”. I’m partial to it because it was the first full-length novel I ever read entirely in Spanish. Translation: “On the day that they were going to kill him, Santiago Nasar woke at 5:30 in the morning to wait for the boat the bishop would be arriving on.”
“Sing, goddess, the anger of Peleus’ son Achilleus and it’s devastation…”
-The Iliad, Homer.
“Sing to me of the man, Muse, the man of twists and turns, driven time and again off course, once he had plundered the hallowed heights of Troy.”
-The Odyssey, Homer. Neither of these can come anywhere close to capturing the true beauty of the original Greek.
Not fiction, but I feel morally obligated to add it anyway:
“Can you tell me, O Socrates, can virtue be taught?”
-Meno, by Plato. Perhaps the most wonderful Platonic dialogue in existance.
“He was one hundred and seventy days dying and not yet dead.” The Stars My Destination, Alfred Bester
I, Tiberius Claudius Drusus Nero Germanicus This-that-and-the-other (for I shall not trouble you yet with all my titles) who was once, and not so long ago either, known to my friends and relatives and associates as “Claudius the Idiot”, or “That Claudius”, or “Claudius the Stammerer”, or “Clau-Clau-Claudius” or at best as “Poor Uncle Claudius”, am now about to write this strange history of my life; starting from my earliest childhood and continuing year by year until I reach the fateful point of change where, some eight years ago, at the age of fifty-one, I suddenly found myself caught in what I may call the "golden predicament from which I have never since become disentangled.
Robert Graves, I, Claudius