The closest I ever came  and that unwittingly  to a brush with algebraic geometry, was in freshman calculus. (This was back in the ‘sixties  before your time.) The instructor was Robin Hartshorne, a young and winsome elf of a man. He was an engaging lecturer, teaching from or at least in parallel to a beguiling text (Spivak’s
Calculus, so utterly different in spirit from the dry Thomas treatise that had repelled me in high school to the point of dropping the class); moreover he was  now that I think back on it  the first deeply intelligent teacher I’d ever had (though there were to be others  notably
Gleason): up through high school, there had been no hint that such creatures even existed.
Still, he wore his learning lightly, like his tweeds. The class was fun. Particularly endearing  though also startling, at the time  was one day in the second semester, when he was presenting the topic of definite integrals  painful but necessary, rather like a rectal exam. He was chalking away, when all at once he seized up, staring in bemusement at the blackboard; then turned to us with a sheepish grin.
“It’s been a long time since I’ve done one of these,” he said.
~
Now  if the anecdote stopped there, it would be just one more ultimately stupid instance of Genius Porn : the populace cooing contentedly when told that Einstein flunked gradeschool math, or that Gauss was late to learn to speak, or that Erdös tried to cut a grapefruit with a butterknife (which indeed he did, though it was a craftily calculated move). These falsely flatter our vanity. They are the opposite of a much better genre of joke, which you need a bit of math to actually understand, such as the one about von Neumann and the summation of infinite series.
For the incident, trivial until viewed in the light of later developments, did there and then plant the seed of doubt and wonder, as I sat theretofore clueless in the second row. His being momentarily at a loss struck me at the time as quite surprising, almost inexplicable: as though a testpilot, stepping from his aircraft into his roadster, were to stare at the ignition and say, “Remind me how to start one of these.”
Had I continued with a chemistry major as originally planned (well,
originallyoriginally an English major, until I realized, with chill horror, the error of my ways), and thus retreated or perhaps advanced depending on how you look at it, into evermoretechnical intricacies and mechanical practicalities, the significance of that incident would never have become apparent. But as it was, Hartshorne’s class (and Spivak’s sparkle) were instrumental in turning me towards a concentration in pure mathematics  which was the only kind of mathematics they really taught at Harvard (golden memories of that climate of abstraction
here), and eventually towards having a go at Berkeley towards a Ph.D. Accordingly I was to be introduced to asyetunsuspected levels of intellectual depth, such that each, compared with the one before (I speak loosely; they are basically incomparable), is as the definite integral to 2 + 2:
rising like the serried ranks of angels. And though I myself never progressed beyond the level of the cherubim, it was sufficient to glimpse that empyrean wherein, indeed, you might forget the particular monkeytricks used to solve thorny individual definite integrals (basically you just memorize these, storing them for reference like tools in a toolkit, unless you’re Euler or von Neumann, in which case you rederive them instantly from scratch, or simply perform a bruteforce numerical calculation in your head).

As each glowing level is added, the one below becomes obsolete ... 
Moreover, the tired old carthorse of the calculus was, it turns out, very far from the centers of Hartshorne’s research interests, which are almost unimaginably abstract. In that pokey little classroom in Massachusetts, he was really only on loan to us from Sagittarius, having once studied with Grothendieck, a confirmed extraterrestrial. It was bruited about that he had something to do with something called projective geometry; but only much later was I to learn that he is one of the pioneers of …
sheaf theory
… a topic so ferociously abstract, that even to define what sheaves are is utterly beyond me. (Wikipedia doesn’t even try, observing that “their correct definition is rather technical”.) Nay, wert thou to gaze upon this theory naked  not even to speak, not even to breathe of its further generalizations in topos theory  ‘twould make thine eyes, like stars, to start from their spheres, and thine each particular hair  nay more, thou wouldst in sooth explode in flames, as did dame Semele, when all unheeding she beheld, unveiled, great Zeus in all his lightning !
~
~ Posthumous Endorsement ~
"If I were alive
today, and in the mood for a mystery,
this is what I'd be
reading: "
(Je m'appelle Evariste Galois,
and I approved this message.)
~
~
~
~
As so often when some movement of math has been seen streaking off westwards out into the void, presumably never to been seen again by mortal man, it reappears shining in the east, reborn in some applicable form  thus suggesting, you will notice, that the global topology of the noösphere is toroidal. In the present case, algebraic topology has come to be crucial in such applications as: string theory (via its prior discovery of CalabiYau manifolds), coding theory, cryptography and steganography  which means that the juicy bits are probably highly classified.
And this raises the interesting paradox, which Epimenides would have relished, whether someone like Hartshorne is Cleared for the contents of his own head.

Robin Hartshorne as seen in a recent spectral
image. Since the old days, he
seems to have sprouted quite a bundle of fibres over his basespace. 
~
I recently happened upon an essay by the usually abstruse Samuel Eilenberg (of
EilenbergandSteenrod notoriety), written for a collection sponsored by the Office of Naval Research. In deference, perhaps, to the needs of our seamen, Professor Eilenberg permits himself some observations and analogies that lie within the reach of the common folk. Thus:
The analogy between sheaves and covering spaces is very close.
 Samuel Eilenberg, “Algebraic Topology”, in: T. L. Saaty, ed. Lectures on Modern Mathematics, vol. I (1963), p. 110
Bingo !
Coveringspaces I get. Of course, I don’t actually see the analogy, but at least it’s a comfort, knowing that there is one.
[Update, Thanksgiving 2014] Edward Frenkel, noticing my perplexity, kindly sent in
this explanation:
Coverings and sheaves are related. And
it's not just an analogy. A covering space is an example of a sheaf (the
simplest example): It is a sheaf of finite sets (provided that the covering is
finite).
Namely, given a covering p: C > X
of a manifold X, and given an open subset U of X, the set of sections of of the
corresponding sheaf over U is just p^{1}(U) [the preimage of U in C under p].
Note that the stalk of this sheaf over a point x of X is just the fiber over x
[the set of points in C, which project down onto x under p; p^{1}(x)].
The covering gives us a way to
"glue" these fibers together (indeed, settheoretically, C is the
union of these fibers  but it's more than that, because C is a manifold, just
like X; so C is not a "disjoint" union of these fibers, they are
really "glued" together in a particular way).
The simplest covering space is the
trivial one: a union of N copies of X, each mapping identically to X under p.
Here is a nontrivial example: let X be
a circle. Now take the Moebius strip in which this circle is the circle
"in the middle." Take the "edge" of the strip  this will
be your C. Notice that for each point in your original circle X, there are two
points in C. But C is NOT the union of two circles (which would be the trivial
double covering). In fact, C is just ONE circle, covering another circle (our
X) in a nontrivial fashion.
A general sheaf is very similar. The
difference is that the fibers could be infinite, or they could be vector
spaces, etc. But the idea is the same.
~
We earlier discussed the memoir
Souvenirs d’Apprentissage, by a pioneer of algebraic geometry, André Weil. Avid for more, we got hold of a copy of the memoir
Random Curves (2008), by a contemporary algebraic geometer, Neal Koblitz, well known to anyone with an interest in Elliptic Curve Cryptography. (It came in via InterLibrary Loan  interestingly, the lending institution turned out to be the U.S. Naval Academy in Annapolis. Compare the publishing venue of the Eilenberg article referenced immediately above. We salute the broad interests of our midshipmen!)
Both authors have a wide range of interests and experience outside of mathematics; both engaged in extensive foreign travel; and it is of this that they principally write: the reader will enjoy these accounts for their own sake, but we set down the volumes with a twinge of disappointment, that we are no closer to insight about algebraic geometry than we were before. However, one biographical detail did strikingly stand out. Both authors took a principled stand against unjust wars: and this, not simply by penning valiant Letters to the Editor from the safety of their studies, but by a brave and almost reckless defiance while actually serving in the armies of their respective countries: actions that could easily have led to their injury or even death, and which did actually lead to their imprisonment (and, in Koblitz’s case, to a severe beating). But what is truly remarkable is that, in both cases, they used their time in the slammer far more profitably, mathematically, than most of us use ours even in the best of circumstances. It was there that Weil did his seminal work, and there that Koblitz returned in concentrated form to the practice of algebra which he had largely abandoned during two years of political turmoil.
Intrigued, I wrote to the latter author, inquiring whether, from the standpoint of Kolmogorovstyle measuretheoretical probability theory, we may validly generalize from this sample of two (2); and he was kind enough to reply:
I love generalizations based on small samples. I'm sure a lot of algebraic geometry was nursed at the Indiantown Gap army stockade!
Thus encouraged, I here make bold to speculate about the martial philosophy of Robin Hartshorne. I know nothing of his politics, but truly cannot imagine the man wielding an M16. Or a flyswatter, for that matter. Were a mayfly to venture into his office, Hartshorne would no doubt observe the pattern of its flight (musing all the while on the brevity of this earthly life) and calculate whether that trajectory describes an elliptic curve.  Which, come to think of it, it just well might. Many mathematical treasures remain to be unearthed in the field of biology! (For a few of these, see the fine book by Ian Stewart,
Life’s Other Secret.)
~
A rather huffy response to modern algebraic geometry, which
it is a pleasure to reproduce
mainly because I do not understand the subject, and which suggests (like
the characterization of Category Theory back when I was in college, as “the
higher macramé”) that (as with these newfangled things called “computers”) I
am perhaps not missing much:
Attempts to extend the geometry of
secondorder surfaces and the
algebra of quadratic forms to
objects of higher degrees quickly
leads to the detritus of algebraic geometry, with its discouraging hierarchy
of complicated degeneracies, and answers that can be computed only
theoretically.
 Vladimir I. Arnold, Lectures
on Partial Differential Equations (Russian edition 1997; English
translation 2004), Preface.
Harumph! Hear
hear!