If we're going by the literal meaning of the phrase, my "white whales" are Aero Fighters, Hagane: The Final Conflict, and Ninja Gaiden Trilogy (SNES); Ninja Five-O (GBA); The Misadventures of Tron Bonne (PS1); and Gotcha Force (GCN).
From time to time I end up finding one or two priced "reasonably" on eBay, but I have never caved in even though I seem close to. Last year I was about to close the deal with Gotcha Force for $155, but in the end I didn't. I'm stingy when it comes to money, but I've made peace knowing I might bite the bullet with at least one of these someday. Hopefully by then they drop in price.
Hopefully.