The quality or state of being gaspingly out of breath, or the characteristic of being inclined to gasp frequently.
From gaspy (adjective) plus -ness (Old English suffix forming abstract nouns from adjectives). A rare formation in modern English.
Gaspiness is barely used in modern English, but it shows how freely older speakers could create nouns from any adjective—they'd just add -ness and trust readers would understand 'the quality of being gaspy.'
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