The quality or state of being cringe-worthy; the degree to which something causes embarrassment or discomfort.
Cringing plus the noun-forming suffix -ness, which converts adjectives into abstract nouns representing their qualities.
The 'ness' suffix is equally productive as 'ly'—it allows English speakers to instantly convert almost any quality into a noun, which is why you can create words like 'cringingness' and have them be immediately understood, even if they're quite rare.
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