Interpretation of "GRE subject test is required, but your application will be considered anyway" In the US, many PhD programs require applications to submit the results of one of the GRE subject tests. For example, almost every (serious) PhD program in mathematics that I've seen requires it. However, when looking at the admissions webpage of some programs, there sometimes seems to be some kind of grey area when it comes to the subject test.

For example, on Harvard's webpage for the mathematics PhD program, it is written that:


  The Department requires all applicants to submit GRE Mathematics Subject Test scores. Applicants should check on the ETS website for test dates in their area to insure the scores will be submitted before the application deadline.


So far so good, and if they left it at that, the message would seem to be: "you have to submit the GRE subject test, otherwise your application will be incomplete and rejected outright". However, it goes on to say:


  While the admissions committee reviews all applications submitted by the deadline, missing math subject test scores are one less data point available to evaluate the application. Depending on the applicant pool and the strength of the application materials, the missing subject test scores may put the application at a disadvantage.


How should one interpret this? Does it mean:


"We thank you for the $105 donation, but your application lacks critical information and will almost certainly be rejected"; or
"Your application will be considered, provided it somehow demonstrates that you have a sufficiently strong mathematics background to succeed in our program"?


One one hand, it seems possible that they would want to "consider" applications without the GRE subject test, in that they receive the admission fee and that they can discard the application with very little time and effort (after all, they said the GRE was necessary). On the other hand, it seems silly that Harvard should reject an applicant who, by all accounts, seems to be this generation's Terence Tao, for the simple reason that he did not take the GRE subject test.
This is certainly poorly and confusingly worded, but I think I can explain the reasoning based on similar issues I've seen with other departments.  Here's my interpretation:

The department expects every applicant to take the GRE subject test.  However, they know that every year a small fraction of the applicants screw up and don't manage to register in time to take it in the fall.  It's impossible to take the test at the last minute, so at that point there's no way to fix the situation.  The department doesn't want to automatically reject these applicants, for exactly the reason you mention (what if someone's the next Terry Tao?), and they are worried that if they list the GRE as an absolute requirement, then the university administration might force them to do that.  On the other hand, they also don't intend to offer applicants the discretion to decide for themselves whether taking the subject test would strengthen their applications.  They want everyone to take it, with the possibility of forgiveness for outstanding candidates who screw up.  These goals get combined into the confusing statement that it's mandatory but your application can still be considered without it.

I don't know how the Harvard math department evaluates applicants with no GRE scores, but I can tell you how my department does.  We're a little happier if we see an apologetic note about how the applicant intended to take the test but failed to do so in time, but of course we know such a note proves nothing.  Our default assumption is that the applicant was worried about doing poorly, so we start asking questions like "How badly would we guess they might have done?" and "Would we have accepted them anyway, even with low GRE scores?"  If the application is otherwise fantastic, then the chances of acceptance remain good.  However, these are the cases in which we would have cheerfully made excuses for low scores (maybe the applicant just isn't good at standardized tests or had a bad day), so nothing is gained by skipping the exam.  An otherwise marginal applicant with no GRE subject test score will be rejected, and a strong but not fantastic applicant will be at a disadvantage.

So my best guess as to what the policy means in practice is something like "if you don't take the exam, we'll be annoyed at your irresponsibility and we'll evaluate your application with a presumption that you wouldn't have done well on the exam, but we'll still consider admitting you."  This means you have a strong incentive to take the exam if you can.


  On one hand, it seems possible that they would want to "consider" applications without the GRE subject test, in that they receive the admission fee and that they can discard the application with very little time and effort 


In a typical U.S. department, application requirements and graduate admissions decisions are handled pretty much entirely separately from application fees.  For example, many people on the committee won't even know offhand what the current application fee is or where the money goes within the university.  In particular, whoever wrote the policy about GRE scores probably wasn't thinking about application fees at all.