This question is prompted by the fact that I was discussing this with friends last night. The books I would have, at easy reach, for my hypothetical children (who would probably have to be girls for my list) who love to read as much as I do would be:
-All of the "Little House on the Praire" books.
-All of the "Anne of Green Gables" books.
-Little Women
-Walk Two Moons by Sharon Creech
-The Giver by Lois Lowry
-Madeleine L'Engle's "Time Quartet" (A Wrinkle in Time, A Wind in the Door, ...)
-A Little Princess
-A Secret Garden
-A handful of the 1950s Nancy Drew books
-All of the Narnia books
-All of the Harry Potters
-A handful of "The Boxcar Children"
-Harriet the Spy
-Anything and everything by Roald Dahl
-The Phantom Tollbooth
-Shel Silverstein's poems
..

This list is basically what I would buy for myself if I had the funds.

You know, I have a good friend who wanted to do a film treatment for The Phantom Tollbooth, so I tried reading it a bit over a year ago and wasn't engaged at all. I had to return it to the library in the end, because I ran out of time to attempt to finish it. Maybe I'll try again sometime.

As to your question, it's a very interesting one, particularly given that you are clearly taking into account a wide range of your children's ages, which makes it much more complicated.

Well, to start I'd have to have each of the books in the logo and banner:
Dominic (possibly my favorite children's book ever)
Abel's Island
The Going to Bed Book
the entire The Church Mouse series
and The Isle of the Skog.

To that I would add (NOT pictured above):
Pajama Time! (so much fun to sing!)
all of the "Oz" books
the "Little House" books
The Jungle Book and Just So Stories
The Prisoner of Zenda and The Three Musketeers
Treasure Island (so much fun to read aloud!) - oh, face it, I'd want a complete Stevenson collection
The Hobbit, or There and Back Again
The Ear, the Eye and the Arm
the complete "Tintin" collection
Tales of the Arabian Nights
Complete Works of William Shakespeare
The Lord of the Rings
Tales of a Fourth Grade Nothing
A Series of Unfortunate Events
The Complete Adventures of Sherlock Holmes
the Story of Ferdinand
Brighty of the Grand Canyon
White Fang and Call of the Wild
The Black Pearl
Bambi
When We Were Young
complete sets of Brothers Grimm, Hans Christian Andersen, Cuchulain stories and Mother Goose
The Cricket in Times Square
the "tripods" series

Oh, I can feel the list going on and on. It's nice, though, since I already do have almost all of these (not Bambi and I think not Ferdinand anymore).

Cressida, what if your hypothetical children were boys?? :)

I couldn't get my brothers into Little Women or Nancy Drew, but they did love Jon Scieska's "The Stinky Cheese Man". When they were pre-readers and early readers, they loved anything by Ezra Jack Keats or Maurice Sendak. My littlest brother also like Beatrix Potter's "Benjamin Bunny" and "Peter Rabbit" when he was a wee one.

I know we had Kipling's "Just So Stories", but my family also had a great collection of other animal stories and folktale books. Need to find those.

Great picks from both posts!