I know people are just trying to be polite and respectful, to not pile more on me when I've already got so much piled on me. The intent is good. The execution isn't. Invariably, "are you busy?" just makes whatever they want take more of my time.
The problem is, if they mean "are you doing nothing at all" the answer is always "no". But what they really mean is "do you have enough time to help me with this thing I have in mind". Trouble is, this thing could range from "answer a quick yes-or-no question" to any of various things that could take hours. It could also involve things that I can do right now while I'm doing whatever else I'm doing, or it could be something that requires me to stop what I'm doing or even go to a different place, perhaps a place I am not able to go right now even if I did have time.
I tried various answers that made clear the dilemma, including just asking "busy for what?" but they all just confuse people half the time (requiring more time be spent on meta stuff before we can get to the actual issue) and put people off the other half the time (they take all those answers as "yes, I'm busy, and rude besides" and react rudely in response). In the end, any response that attempts in any way to address the fact that the question doesn't give me enough information to answer it, always comes off as me being the one not giving enough information, and/or being a jerk.
"Try me" is suitably vague -- as vague as the original question, and in the same way -- yet it manages somehow to be taken, most of the time, as "yes" enough for them to get to the point. (The point being telling me what they're actually asking, at which point I can give the real answer about whether I can spare the time for it.)
The logical systems analyst in me can't help but note that there's nothing whatsoever in this exchange ("are you busy?" "try me."
If people really wanted to be polite, what they would ask is a very brief version of their question that still had enough in it to let the person answer whether they're too busy for it. But that's never going to happen. That said, if you happen to be asking me, know that I won't mind if your question is fully formed. Quite the contrary. It'll probably save us both time. And if I seem irked (because you hit me at a really busy moment, maybe) it's not because of how you asked: if you'd only said "are you busy?" I would simply have been even more irked.