There are some "unpaired" un- words, like "ungainly". This now means "clumsy/awkward in movement". This derives from an original word "gein", which meant "suitable for purpose, helpful, reliable". The base word "gein" has gone out of use in English, and the sense of "ungainly" has shifted, from "inconvenient, unfit for purpose" to its modern meaning. The shift in meaning was possible because it was an unpaired word.
A similar sense shift happened with uncanny. The word "canny" dropped out of English (as spoken in the South of England) and the unpaired word "uncanny" shifted in meaning. However the word "canny" was preserved in Scottish dialect, and then re-entered English from Scots. It is still much more common in Scottish English.
Another similar pair is "untoward" It is not an antonym of "toward". But means "unexpected and inappropriate or inconvenient."
I've thought of another one: Unwind, means "relax", and while we might say "I'm wound up", we don't use "wind" to mean become stressed.