Everyone is doing it (as Tyler Cowen invites, Julian Sanchez explains and Andrew Sullivan anthologizes). No particular order, but maybe more or less as I encountered them...I'm also limiting myself to one per author (almost), but in most of these cases it's this one, plus everything else.
1. Lewis Carroll, Through the Looking-Glass and What Alice Found There. Logic, whimsy, words, nonsense, philosophy, comedy. There must be a dozen bits in it that I find constantly in whatever I do.
2. The Tanakh (that's Torah, Prophets, and the other Writings, for those with different bibles). Stories are not just for enjoying, or for learning simple lessons from (not my brother's keeper? duh.), but they are also texts for endlessly analyzing, interpreting, and savoring.
3. Bill James, The Bill James Baseball Abstract 1983. His second mass-market Abstract, and the first I read. Questions should be substantively important, and interesting. Empirical questions can be answered by analyzing evidence. Write well; write well about quantitative evidence. Credentialed experts can be totally wrong. (The HOF book is underrated).
4. Nelson W. Polsby, Consequences of Party Reform. I was a Polsby student in grad school. I miss him a lot. The lessons? See Bill James, above. Really -- although far more systematically. Substantively, ignore most of the particulars (which in my view are now dated) but pick up on wonderful stuff about the nature of political parties and American politics. A terrific writer, too. (In addition to his books and articles, his textbooky efforts -- Presidential Elections, and Congress & the Presidency -- have terrific stuff).
5. Richard Neustadt, Presidential Power. OK, not every great book is well written. But: "the Constitutional Convention of 1787 is supposed to have created a government of 'separated powers.' It did nothing of the sort. Rather, it created a government of separated institutions sharing powers." Really understand that, and you'll be in pretty good shape in understanding American politics.
6. Garry Wills, Nixon Agonisties. Beyond the specific stories, which he researches and tells so well, something that I couldn't explain about the interaction between culture, ideas, and politicians comes through when reading Wills. And another wonderful writer; his retelling of Checkers is a classic. (The Reagan and Washington books are my other two favorites).
7. William Shakespeare, Henry V. Or anything else, really, doesn't matter; it's Shakespeare, after all. As far as influencing me, probably the main thing would be that fiction -- especially plays, movies, TV -- can teach things about politics that we can't get to in other ways. I hear there's stuff beyond politics in Shakespeare, too. Yeah, I know, it's trite -- Shakespeare and the bible -- but fine, call me trite.
8. Richard Fenno, When Incumbency Fails. My real introduction to thinking seriously about representation. Which, regular readers know, remains an obsession. Plus I just admire anyone will to do what he does to collect data. (All his home style books are essential).
9. Hanna Pitkin, Fortune is a Woman. More about words. Don't accept assumptions, about anything -- figure out what your assumptions are, and follow what happens if you suspend them. Read all people fairly (except perhaps J.S. Mill), assuming that they have something to say, even if their assumptions trip them up. Another outstanding writer, and brilliant thinker and teacher. (Everything she's written is terrific). Oh, and this is also a way to sneak Machiavelli onto the list.
10. Hannah Arendt, On Revolution and The Human Condition. And everything else, but I couldn't narrow it to just one. I don't really know how to boil this down to a sentence or two...for me, at least, she is essential.
And add a wildcard. I'll mention Baum, Asimov, Heinlein, and Lloyd Alexander (yeah, I know it's not just politics, but that too); Duane Decker (and his Blue Sox heroes), John Tunis, and Roger Angel (think about baseball all the time); Andy Beyer, Tom Ainslie, Donald Sobol (solve puzzles!); Gordon Wood, Bernard Bailyn, and James Madison (that USA place is sorta interesting); and a dozen or two political theorists from Plato and Plutarch on.
(Edited, to correct for my inability to count to ten)