inquisitiveness

The quality or state of being inquisitive; the disposition to seek explanation and information; curiosity to learn what is unknown; esp., uncontrolled and impertinent curiosity.

Mr. Boswell, whose inquisitiveness is seconded by great activity, scrambled in at a high window.

Curiosity in children nature has provided, to remove that ignorance they were born with; which, without this busy inquisitiveness, will make them dull.

noun

1. Undue interest in the affairs of others: curiosity, curiousness. Informal nosiness, snoopiness. See investigate.
2. Mental acquisitiveness: curiosity, curiousness, interest. Idioms: thirst for knowledge. See investigate.

The noun has one meaning:

Meaning #1: a state of active curiosity
  Synonym: curiousness