So I figured that since I have such a bizarre relationship with the RECAPTCHA feature, I would start a thread to record my adventures for posterity. Please, feel free to pitch in with some of your own! 
This is what sparked the thread: 
 
As you can see, I don't know whether the RECAPTCHA expects greatness of me, if it has needs that only *I* can supply, if it wants to share its love of irrational mathematics with me, or if it's simply insane. 
I'm betting on the latter. 
Update: I fooled the RECAPTCHA with a simple "3/2," so we can rule out the mathematical inclinations.