Sorry to hear that you have to go through this. It’s a shame that the colleges have to turn away some of the best unconventional talents using the standardized testing. I once had a chat with an Ivy League admission officer, who told me that his first step was to weed off those with quantifiable qualification including the test scores lower than the cutoff line. I told him that he would potentially miss certain genius type who was not necessarily well rounded, his answer was: yes, so what, we have more than enough great talents in the pool who happen to be ALSO well rounded. Such is the reality with the admission bureaucracy.

Here are my suggestions for you:

1. There are fortunately many graduate schools that do not require GRE, you may want to check them out.

2. Write to the program director or professor whom you intend to pursue the graduate work with, ask GRE to be waived for you, sometimes when they badly want to have you on board, they may work something out specially for you with the admissions office. Exception is always a possibility, granted or not,  it does not hurt to ask.

 3. The math portion of GRE weighs 1/3 of the total if I remember it correctly, I wouldn’t recommend to skip it, doing so might drag you below the cutoff line even if you score perfectly for the rest. It it turns out that GRE is unavoidable, please know that it’s actually not too bad, except that putting up with being assessed is understandably ego-hurting for an highly accomplished person, if that’s being the case, don’t forget that you could get your revenge soon enough by your display of academic brilliance upon entering the graduate school. 

Best Wishes and Good Luck! 

Dear Il'ja, if you've mastered the intricacies of Greek and Hebrew grammar, the GRE should not be cause for worry. From what I remember it's a moderately predictable test.  This is not to say it won't require some regular work, as if you needed to train to run two miles in twenty minutes in order to be admitted to graduate school, or to take a particularly arcane driver's license exam. But certainly for you it should be doable.
There is a specific range of skills being tested, which seem strange at first but quickly fall into patterns; and there is an entire cottage industry of GRE workbooks aimed at explaining them. You might consider it a cultural experiment -- buy yourself two or three of those workbooks, and try to understand from this strange hybrid of computation and theory what purpose mathematics is thought to serve in the modern world. (Would be interested to hear your conclusions...)
So as to lighten the mood a little, you might want to acquire some more inspiring mathematical books too -- tales about Paul Erdos for instance, or the "Proofs Without Words"  Emily suggested once.

On the other hand:
Not having not sat on admissions committees myself, I don't know whether it is possible to waive the exam as Qian suggests but it's an interesting possibility and worth looking into. Here (http://www.ets.org/research/policy_research_reports/rr-83-20) is a study, albeit from 1983, done by the testing agency which basically concludes that there are nontrivial issues with the test's ability to gauge skill of older test takers. I read their blandly worded "A review of the literature on learning ability and age suggests that methodological problems and changing times probably account for most of the reported age differences in both learning ability and performance" as telling us that the test is really a product of its specific generation, culture and time. I imagine an inquiry would not be out of place, perhaps directly discussed with the department rather than the GSAS. [Does anyone have advice about what specifically to say?]
Keep in mind though that, more and more these days, there may be external factors having to do with general administrative bookkeeping or accreditation which the department may be powerless to bypass. Still, they may be able to assure you informally that the score won't be the deciding factor. Contact of this sort, when done well, and even if it does not have any effect in waiving the test, may turn out to be a nice excuse to interact with the department and thus to increase their interest in you.

In sum, I'll repeat my vote of confidence. The main issue is simply that it would save you a fair amount of time to avoid the aforementioned cultural experiment, rather than any inherent difficulty in the test itself. 

Molly and Qian,

Credit is due.  Your practicality is outweighed only by your generosity of spirit and your encouragement.  I came looking for Socrates and I got Barnabas along in the deal!

It would never have occurred to me to request to opt out of the test.  I'm skeptical, frankly, not so much of the department's goodwill, but of my academic record, given its vintage.  My Ukrainian grad work, per se, is, I've learned, dismissed as irrelevant.  Interestingly and conversely,  the recommendations of those writers , publishers, and academics with whom I work/study here are considered to be hugely relevant.  Go figure.

It's hard to see them agreeing to an opt out, but - for all the reasons you mention - it still might not hurt to ask.  This is a bit of spiritual conundrum for me.  When I was active in the parish, it was a challenge to carry a loving spirit and an open mind into the confessional some days: to discard the politics, the potential manipulations and any thoughts that might abridge humility when dealing with the penitent.  That's prologue to the reality that, while I want to give the committee the opportunity to do the loving thing, I realize that they are still just a committee.  They have policies.  They undoubtedly have state/federally-mandated admittance protocols.  Someone might come to the task of ticking the boxes on my application fresh from a jawthrusting argument with a husband, landlord, or department head.  They might ordinarily be favorably disposed toward an older guy who asks "cut me a break here", or they might hold onto that little nugget.  When my test comes through and my inability to compete with the kid from India, Taiwan, or Tulsa is made painfully, quantifiably, evident...well, then, I can imagine "isn't this the guy who wanted to opt out of the GRE?  Aha!" shaping the narrative.

The math is the math is the math.  I've already got my hands on the "Princeton" and "Barron" guides, and Qian, you are spot on with your comments about ego.  The manuals read like something written by a 25-year-old IT-specializing grad student.  Still, I begin working through it, my dentures (ok, I don't wear dentures), at a slow grind, thoughts of violence toward whippersnappers and dread for the future of the planet flooding my brain pan.  And I discover in the process that I can learn things from people who grasp, and likely use, algebra far better and more than I ever will.  I feel like a kid again just going through the mechanics of "solving for x".  I suppose the only question remaining is to uncover an appropriate strategy for carrying along into the testing room the inchoative stages of calm I'm experiencing now at the feet of my "tutors".  I've also located the "Proofs Without Words" I&II, and an Erdos collection of essays, Molly, though I may save the latter until after the test.  When I'm feeling smug.

It also seems like the "assuring [me] informally" process has already begun.  Russian, Ukrainian, and Slovak fluency count for something, and I was advised to lean heavily on these in my statement of purpose.  The woman I'm corresponding with even feels comfortable enough at this point to make small, dry jokes.

My dread diminishes.  An friend from the culture sphere told me last evening that I'm not "scrunching" my forehead so much.  She'd just picked up the word 'scrunch' and was excited to use it.

Cognizant of how much I sound like a ballplayer giving an interview after a big game, I'm no master of Greek.  I am glad to be in the stable of those who try to pull that plow through the field on occasion.  What I know is that I've had some outstanding teachers.  And encouragers.  I'm grateful to you.

So, old dog, old tricks training has begun in earnest.  I'll try to share any insight I gather, re materials, strategies, etc., as I go along.  The fear is still there, but the will is waking up.

peace,

Il'ja