Basically your Moby Dick, the one that you get close to but never able to get your hands on. I currently don't have one but I do have one for a bar, I've nearly been able to get a globe bar a few times but never able to seal the deal.
My white whale pipe would have to be a collection actually. The Peterson Return of Sherlock Holmes set. I'm wanting to buy the entire set all at once with the pipe rack, but I can't seem to save up the money for it.
Nifty responses, fellows.
Keep them coming.
I'd submit that to be a "white whale," the pipe must not only be finest quality or expensive or even your perfect person taste, but also in some way difficult to obtain: such as a very particular estate like I've defined, or a war era Dunhill (see Neill's nifty post: A Wartime Treasure), or any Poul Ilsted pipe (as I understand he'll only sell you a pipe if he likes you).