This is a relatively simple one that has been around for quite a while. “A horse, a horse, my kingdom for a horse!” speaks the eponymous King Richard in desperation as the battle turns against him in William Shakespeare’s c. 1591 Richard III.
Modern usage of this snowclone is much more lighthearted–or so I judge from the variations of X, which include book, nap, glass of milk, whore, cake, couch, goat, calf, pun, loaf, cough drop, blu-ray burner, carder, and swashbuckler’s life, among many many others. The speaker may not have his X at the moment, but
So X is most commonly a physical object, to parallel the original horse, but it doesn’t have to be (nap, pun, brogue). I believe this snowclone is still closely tied to its origins, which is to say that users know it comes from Shakespeare, and that the original X is horse. A possible limitation on X is that it start with a consonant sound–so that we don’t get “my kingdom for an airplane”. All of my top results from a Google search and a snowclone.pl search (which may or may not be enough information) confirm this. There doesn’t seem to be a limitation on the number of syllables X may have, even though the original would have been constrained by the iambic-pentametric form of the play’s verse.
This snowclone also appears in German (“mein Königreich für ein X”), with similar variation. Xs include Ladegerät (charger), Zimmer (room), Sonnenbrille (pair of sunglasses), Brücke (bridge), and Nacht (night). German speakers also generally know it comes from Shakespeare, and seem to place the consonantal constraint on the beginning of X.