Flipping idly through my copy of Kunchū Zenrin kushū 訓註禅林句集改訂版 ("Annotated 'Collection of phrases from the forest of Zen': Revised Edition"), ed. Shibayama Zenkei 柴山全慶, when I found the word 驢年. The characters literally mean "donkey" and "year", and Shibayama glosses it thusly:

In the Twelve Branches system, there is no Year of the Donkey. This phrase therefore refers to a year that will never arrive. Similar to the colloquialism "Year of the Cat, first year of Hōraku" [猫の年、法楽元年]."

I have not located any evidence for that colloquialism, but okay: "Year of the Donkey" meaning a time that will never actually come. I guess that to someone to whom the Twelve Branches were second nature, "the Year of the Donkey" sounds as ridiculous as "when pigs fly."

But if you look up 驢年 in the Nihon Kokugo Daijiten 日本国語大辞典, you find the word 驢年犬日 ("donkey, year, dog, day"), with this definition and a citation from 1771:

Something without value, as the years lived by a donkey and the days passed by a dog. Inferior writing. "The braying of donkeys, the barking of dogs" (驢鳴犬吠).

... Which obviously is quite different.

Then I found this essay by Yoshizawa Katsuhiro 芳澤勝弘, at the International Research Institute for Zen Buddhism, in which the two definitions both appear. According to Yoshizawa, Iriya Yoshitaka 入谷義高 and Koga Hidehiko 古賀英彦, in their Zengo jiten 禅語辞典 ("Dictionary of Zen vocabulary"), quote Mujaku Dōchū 無著道忠 giving the "Twelve Branches" explanation above, but rejects this as "unconvincing" because it fails to explain why the donkey specifically should have been chosen as the never-never animal. They then give a "Worthlessly spent life" definition similar to the one from the NKD above.

But Yoshizawa then offers half a dozen examples from the canon where 驢年 is paired with a word/morpheme meaning "arrive at" or "reach", meaning that it is understood as a point in time rather than a way to spend time.

So it seems that the original meaning of 驢年, at least in the Zen context, is "time that will never arrive." On the other hand, the NKD's 1771 citation for a "time without value"-related meaning shows that that interpretation has at least a few centuries behind it, too. I wonder if the two meanings diverged from a single source or were coined separately.