*If, that is, you're in the category of "most people I run into."
All of these were published in the 20th century, or just after. I'm not well-read enough to imagine I know about more great classic books than you probably do. But these -- while I'm guessing you've heard of most, I have the feeling that for most readers at least several here will be unknown territory. No links -- you can decide where to buy 'em. In no particular order:
Michel Faber, Under the Skin: Not so much for the (wonderful) creepitude, or the various levels of social satire, as for the fascinating mental exercise of imagining what kind of place Isserly is from.
C.P. Snow, The Masters: A shamefully neglected writer, at least in the U.S. The quietly-carried-out election of a new "master" (something between a Dean of faculty and a president of a college) at a midcentury Oxbridge college turns out to be utterly riveting. Also good, from the same series: The Affair.
Tom Carson, Gilligan's Wake: This is a bit unfair as a choice, as I've become slightly acquainted, at least via email, with the guy. But this is a novel that has both heroin-shooting with Daisy Buchanan and Russell Johnson morphing into Godzilla in D.C. What have you got to lose?
Ishmael Reed, Mumbo-Jumbo: Something of a jazz novel, something of an extended dream-poem narrative, with experiments in typography, something of a conspiracy theory about the forces of repression wanting to quash the viral growth of ragtime and black culture in general. Funny and hallucinatory but always retains a curious intensity and focus. One of the finds that made grad school absolutely fucking worth it.
Donald E. Westlake, God Save the Mark: Picked more as an example of Westlake's nearly-unique way with the comic crime novel than as a surefire vote for best-of-his-oevre. As a thirteen-year-old hiding out in his local library from the emotional wasteland of junior high, I found Westlake, like Douglas Adams, a salve for the the soul. This is the one I started with, and why shouldn't you?
Allan Hollinghurst, The Swimming Pool Library: There are not that many books about which one could say both "It's sort of wall-to-wall sex," really, and "it's exquisitely written."
John Ellis, The Social History of the Machine Gun: Gets a bit technical, but it's brief and delivers absolutely what it promises. Why automatic weapons and brutal colonialism go hand in hand; or hand-on-trigger.
Antonia White, Beyond the Glass: A treatment of love and madness that will stay with you. Another British writer who's not well-known enough in the U.S.
Robert Anton Wilson and Robert Shea, The Illuminatus Trilogy: Silly, overwrought, too long, and of mixed quality in terms of its prose, humor, and level of philosophical insight. Half a prank and half a stoned ramble. Still a jaw-droppingly effective transmission of almost everything good about mysticism and esoteric culture, delivered with humility, outrage, and a profound sense of humor about itself. Bonus: makes Ayn Rand laughable on a permanent basis.
David Quammen, Monster of God: A book about the big carnivores that still walk the earth (bears, lions, crocs, and tigers, to be precise), and (more importantly), their relationships with the peoples who live in regular contact with them, and who in many cases are among the world's poorest.
There. Now it's your turn. Please supply at least five in the comments (or elsewhere, and give us a link).