a degree that has run past the normal scale: surpassing greatness, or, as a phrase, beyond all measure
The word is built on the verb "to surpass" (ὑπερβάλλω), whose picture is a throw that lands past the marker. The noun names the amount by which something runs past the normal scale. Greek writers used it for whatever sat beyond due measure, and philosophers used it for the fault of having too much of a good thing. It is also where English gets the word "hyperbole," the figure of speech that overstates on purpose.
Paul is the only New Testament writer who uses it, and he uses it in two distinct ways. Sometimes it is a plain noun with a thing attached to it: the surpassing greatness of the power at work through fragile people, the sheer scale of the revelations he was shown. More often it works as a set phrase, "beyond measure," which names no quality of its own but stretches whatever is being said past any ordinary reckoning.
That stretching runs in both directions. It makes sin utterly sinful, it makes an ordeal in Asia heavier than the strength available to carry it, and it makes Paul's own campaign against the church relentless past all restraint. It equally marks the way of love that outclasses every spiritual gift, and the glory being prepared for people who are suffering. So the word carries no moral charge of its own. It says only that measurement has run out. Nearby words divide the work differently: a surplus of something countable is περισσεία, and standing higher than others in rank is ὑπεροχή.
The boldest use is in 2 Corinthians 4:17, where Paul sets the word twice in a row, one "beyond measure" running straight into another. Nothing else in Greek quite matches it. He is weighing present trouble against coming glory, and the doubled phrase is his way of saying that the scales will not hold.