Why did the chicken cross the road? To get to the other side.
In order to really get this joke, you have to have a certain basic understanding of irony that I know I didn’t have until I was, oh, 10 years old or so. Of course, by that point, I had heard it a million times, so if someone had told it to me, it still wouldn’t have been funny. Has anyone ever actually heard the joke, recognized it as a joke, and got it, the first time they were told it?
Now that’s an original post!
Now that I think about the joke, it sounds sort of Zen, doesn’t it?
I sure hope so. I didn’t bother to search the archives. 
Funny, but I don’t remember ever NOT “getting it”. My memory as a youngster is sketchy at best, so I can’t confirm this, and yes, after hearing the joke repeated millions of times it’s hard to single out “the first time,” but I don’t recall ever thinking this was obtuse in any way.
The “joke” is the same as another joke I’ve heard more than a few times:
Q. Why do firemen wear red suspenders?
A. To keep their pants up!
Same exact concept. I kind of think I “got” this joke as well the first time I heard it. The difference between the two is that the suspenders joke makes the “joke” more obvious. When you first hear the question, you interpret it as asking why firemen choose to wear red suspenders as opposed to any other color of suspenders. The answer is funny because it demonstrates that you were overthinking or misinterpreting the question, which in actuality was why do they wear any kind of suspenders at all. It’s a trick question, and the answer points that out.
The chicken joke does the same thing, just not as obviously. You interpret the question as asking for the motivation of the chicken in choosing to cross the road. Instead, the answer is the obvious, simple reason as to the mechanical result of road-crossing, which is immediately understood to be true (and so obvious that it would never occur to anyone that this was the true basis of the original inquiry). It’s this little twist that makes it funny. Or perhaps demonstrating that the answerer is an idiot… that’s what makes it funny (at least to second graders).
This joke is so universal that many jokes are entirely based on everyone understanding the original. For example:
Q. Why did the chicken cross the road?
A. I dunno… why did the chicken cross the road? (or to get to the other side, either way works with this one).
Q. To get the [insert name of you home-town newspaper here].
Q. Get it?
A. No…
Q. Well, it’s only $3.95 for a month! You oughta start getting it!
or one of my favorites:
Q. Why did the punk rocker cross the road?
A. He was stapled to the chicken!
Then there was Commander Data’s version:
Q. Why did the android cross the road?
A. Because it was carbon-bonded to the chicken!
Why did the Achernar cross the Eridanus?
To get to the other side!
Now that’s humor!
Ok, so maybe not that funny, but it is once you follow This hilarious link.
I don’t care. It’s mine now (see below) (if it’s okay with you, of course). 
richardb, that’s an excellent dissection of the joke. And I really think that the fact that the joke can be analyzed in such detail supports my OP, that the first time I heard it I was way too young to understand it. You know, I probably heard it before I even knew what a joke was, and since it’s a joke on what it is to be a joke, a meta-joke if you will, not knowing what a joke is doesn’t allow me to grasp it.
Of course, all these other jokes we’re throwing out are jokes about a meta-joke, and so are meta-meta-jokes. Thus, if we heard them before hearing the meta-joke that they’re about, we wouldn’t get them. I would even go so far as to say that Dr. Rieux’s is a meta-meta-meta-joke.
Why did the pervert cross the road?
He had his dick stuck in a chicken.
I’m not exactly humor-impaired, but as a kid I never got the typical jokes that we all hear early on in life.
example:
Q: How do you get down from an elephant?
A: You don’t get down from an elephant. You get down from a duck.
So, there I am with my super-literal 5-year-old mind thinking: “OK, but wouldn’t you squish a duck if you sat on it?” By the time I realized it was a play on words, I was in college and was therefore far too sophisticated to appreciate the humor of the joke, which is really too bad since I do find most types of poultry to be inherently funny.
Wow… I’ve never been quoted in a sig before. I’m kind of honored, thanks!
Why didn’t the chicken cross the road?
Because he’s chicken.
Q: And how do you get down from a duck?
A: It’s easy. A duck is not as tall as an elephant.
Yep. college was about the time that I “got” the OP chicken joke and was very disappointed. That’s it? That’s the joke I’ve been baffled by all these years?? I was very literal-minded too (I actually pictured people with combs raking through the dust when I heard “went through the house with a fine-tooth comb”) and still see a black, white and red newspaper when I envision that other joke.
Q: Why did the chicken go to the seance?
A: To get to the other side.
ah aha haha a haha ahah ah ahahaha a ha aha ahaha haha ha aha aha haahhaa ahah aaha aha ha ah aha ahhhhhhhhaah aha
I kill me.
Why did the chewing gum cross the road?
It was stuck to the chicken’s foot, of course.
Because the opossum told him it was safe.