The very strangest one I had, and one which took me quite some time to overthrow, was when the owner of our company demanded that our new product be IE only. If it could work on FireFox, that was OK, but it had to be IE only.
This might not sound too strange, except for one little flaw. All of the software was for a bespoke server software package, running on Linux, and all client boxes that our customer was buying were Linux. Short of trying to figure out how to get Wine (in those days, very unreliable) up and running on all of these boxes and seeing if we could get IE running and training their admins how to debug Wine problems, it simply wasn't possible to meet the owner's request. The problem was that he was doing the Web design and simply didn't know how to make Web sites compliant with FireFox.
It probably won't shock you to know that that our company went bankrupt.
