Every now and then we get a post to meta about ways in which MathOverflow has improved the quality of someone's research. One way I haven't seen mentioned is one that just now happened to me, and (though my memory is a little vague) probably not for the first time:
In the course of trying to devise a homework problem in game theory (!), I stumbled onto an extremely complicated looking expression involving multiple values of multiple polygamma functions. Mathematica told me that my expression, evaluated at any positive integer, is always rational. This was useful for my application, but it appeared to be a sheer miracle. I've spent the better part of the past week trying to demystify that miracle.
Finally I decided it was time to unleash the power of MathOverflow, and carefully composed a question that I then hesitated to post, for fear that I'd missed something obvious. So I spent the better part of the past day going over and over my draft of that question, just to make sure.
And tonight, I had one of those "Aha! It is obvious!" moments. I now understand exactly what's going on. I also see, beyond any shadow of any doubt, that it really should have been obvious days ago, and I have no doubt that if I'd posted it, any number of people would have been quick to see what it took me a week to decipher, and I'd have thanked them profusely, and I'd have felt glad to have my answer, and only a little sheepish. So that would have been a good outcome. But getting it on my own means I had a series of minor insights I will never, ever, ever forget, and that's an even better outcome.
Moreover---and this is really the point---if MathOverflow hadn't been there, I might have decided three days ago that this problem was just too hard for me, and given up on it. Knowing that I could almost surely get my answer by posting ruled out giving up. And then fear of embarrassing myself ruled out posting. The only remaining option was to solve my problem.
So MathOverflow just made me a little smarter by making me feel just a little intimidated about revealing how un-smart I might be (and in fact, in this instance, turned out to be). One more reason to be glad it's there.
Edited to add: Speaking of things that should have been obvious but weren't, I've only just realized that this is not a question and hence perhaps inappropriate. If that's a problem, I won't object to a deletion.