All books by Oliver Sacks. If you haven't read anything by him yet, start with:
The Man Who Mistook His Wife for a Hat and Other Clinical Tales. Sacks, Oliver (That's my number one recommendation ever!)
An Anthropologist on Mars: Seven Paradoxical Tales. Sacks, Oliver
Welcome to Your Brain: Why You Lose Your Car Keys But Never Forget How To Drive and Other Puzzles of Everyday Behavior. Aamodt, Sandra
Mind Hacks: Tips & Tricks for Using Your Brain. Stafford, Tom
The Accidental Mind: How Brain Evolution Has Given Us Love, Memory, Dreams, and God. Linden, David J.
Do No Harm: Stories of Life, Death and Brain Surgery. Marsh, Henry
My Stroke of Insight: A Brain Scientist's Personal Journey. Taylor, Jill Bolte
Brain Cuttings: Fifteen Journeys Through the Mind. Zimmer, Carl
The Tell-Tale Brain: A Neuroscientist's Quest for What Makes Us Human. Ramachandran, V.S.
Cognitive science / Mind / Psychology
The Stuff of Thought: Language as a Window into Human Nature. Pinker, Steven
Why Everyone (Else) Is a Hypocrite: Evolution and the Modular Mind. Kurzban, Robert
The Unpredictable Species. Lieberman, Philip
The Psychopath Test: A Journey Through the Madness Industry. Ronson, Jon
Biology (evolution, game theory, animal behavior):
The Selfish Gene. Dawkins, Richard (must read!)
Animal Behavior. Tinbergen, Nikolaas (beautiful book with great illustrations)
Genome: The Autobiography of a Species In 23 Chapters. Ridley, Matt
A Primate's Memoir: A Neuroscientist's Unconventional Life Among the Baboons. Sapolsky, Robert M.
Curious Naturalists. Tinbergen, Nikolaas
All books by Stephen J. Gould. To begin with:
The Panda's Thumb: More Reflections in Natural History. Gould, Stephen Jay
Maths and physics:
Chaos: Making a New Science. Gleick, James (must read!)
The Drunkard's Walk: How Randomness Rules Our Lives. Mlodinow, Leonard
Naked Statistics: Stripping the Dread from the Data. Wheelan, Charles
Gödel, Escher, Bach: An Eternal Golden Braid. Hofstadter, Douglas R.
The Fabric of the Cosmos. Greene, Brian
The Tao of Physics: An Exploration of the Parallels between Modern Physics and Eastern Mysticism. Capra, Fritjof
PI in the Sky: Counting, Thinking, and Being. Barrow, John D.
Zero: The Biography of a Dangerous Idea. Seife, Charles
The Man Who Loved Only Numbers: The Story of Paul Erdős and the Search for Mathematical Truth. Hoffman, Paul
Best books for premed students
To keep you motivation when you feel down:
Hot Lights, Cold Steel: Life, Death and Sleepless Nights in a Surgeon's First Years. Collins, Michael J.
Complications: A Surgeon's Notes on an Imperfect Science. Gawande, Atul
All books by Oliver Sacks (again). Of more medical-oriented books, the top two would be:
A Leg to Stand On. Sacks, Oliver
Awakenings. Sacks, Oliver
To test your faith (make sure to read these before applying to med school, to check that you really want to be a doctor)
The House of God. Shem, Samuel
Do No Harm: Stories of Life, Death and Brain Surgery. Marsh, Henry
Better: A Surgeon's Notes on Performance. Gawande, Atul
Doctored: The Disillusionment of an American Physician. Jauhar, Sandeep
Fiction and semi-fiction books for neuroscientists
The Curious Incident of the Dog in the Night-Time. Haddon, Mark
Textbooks
Beginner texts
Neuroscience: Exploring the Brain. Bear, Connors, Paradiso. -- Best human-brain-oriented introduction to neuroscience. Also, the only textbook on this topic that reads like a popular book.
An Introduction to Nervous Systems. Greenspan, Ralph J. -- A short, very non-human oriented book with a strong evolutionary tint.
Intermediate texts
From Neuron to Brain: A Cellular and Molecular Approach to the Function of the Nervous System. Nicholls, John G.
Grimoires
Every neuroscientist should have this book, but no one can really master it:
Principles of Neural Science. Kandel, Eric R.
becoming a better writer
Bird by Bird: Some Instructions on Writing and Life. Lamott, Anne
The Sense of Style: The Thinking Person's Guide to Writing in the 21st Century. Pinker, Steven