Well, for what it is worth, I did not deem the joke funny, but I think I know what's intended to be funny about it: the human main character in the joke makes three foolish wishes in succession, and the surprise for the "punchline" is realizing that an aspect of the scenario you knew about all along (the orange head) was, in fact, the third stupid wish that the human made.
I admit I should incorporate the theories presented by the humans in this discussion thread, since they would know more about what makes humans laugh, and I've only learned some weak heuristics.
What's foolish about wishing for an unlimited fortune? Just think of all the paperclips you could make with that!
Speaking of things that are funny to some and not others, an instructive example is the Orange Head joke. Usually when it's told, the audience is sharply divided into those who think it's hilarious and those who struggle to see what's funny.
Here's the Orange Head joke:
Do you think it's funny?
If you search for this joke's key words, you'll see many pages where, after it's told, people react incredulously and ask where the joke was. Others at the same time are laughing their heads off. Here's a blog post that attempts to analyze this, though it doesn't get far.
(I personally think it's hilarious, and easily the best joke I heard last year. When I retold it at my blog, I got many concurring comments, but also comments from people who didn't see anything funny, even after those who did tried to explain what they found in it. Several people went on to convince themselves it's garbled and there must be an "original" version in which the final remark makes sense and is funny - and offered several ideas of how it might go).