Quote from redlinks

You mean to tell us that you've never been annoyed by someone who always asks why all the time?

Only if they are just mindlessly doing it to troll. Meaning that they question has no purpose, substance, or meaning, and isn't a genuine/honest question.
In other words:
It's not about the word "why" (or any other of the options), but the substance of the question, that determines if it is annoying. Just as I said in my initial post.

I wonder why "why?" is so far in the lead (followed by "what?" no less!). For "why?" there's always the response "why not?"

Honestly, "why?" and "what?" are my favourite questions. (Okay, my preferred subjects are showing here, but still.) Yes, "why?" can easily go on indefinitely, but if someone keeps asking it (which is annoying for any question, to be sure), "why not?" usually solves the issue. Truth be told, I find "how?" "where?" and "when?" more annoying, but then again, I am decidedly not mechanically-minded and am pathetic at rote memorization (that, and I find recitation of facts incredibly boring).

Apart from people asking something, anything, incessantly, I'm guessing that which question annoys people most depends on how your mind works / your skill-set. E.g. in my case, questions of how something occurred (beyond the bare bones) and questions of specifics I may or may not remember are far more annoying than questions of quality ("what-ness," otherwise known as quiddity) and questions of causes / reasons (the whys). Someone with the opposite skill-set would probably have the opposite situation, no?

Of course there are also plenty of people who're bothered by the lack of a definitive answer (one of my siblings is a little like that), possibly more than the number of people bothered by the answer being definitive (like me) — well, it isn't a bad thing either way. People who are generally bothered by answering "I don't know," are another matter altogether. Although, sometimes, that situation can make you feel stupid, i.e. when the information you can't recall or don't know is supposed to basic (which is me about a third of the time — what's the word for __ again?).

Of course, in all of this, I'm just speaking generally and of generalizations.

Other than that, I'm in agreement with zarlan.

So many people chose "why?", and I think I understand their reasoning, but I personally chose "how?"

"How?" is simultaneously the most difficult question to answer, and sometimes there isn't even any answer to it, which makes it super annoying.

Even so, it's probably the most important, as understanding how things work in our world can lead to great advances, not only in scientific achievement, but also in an individual's own life.

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Quote from vigorousjammer

and sometimes there isn't even any answer to it, which makes it super annoying.

"I don't know" is a perfectly good answer.