Most inquisitive, having no afternoon rest, almost (7)
I believe the answer is:
nosiest
'most inquisitive' is the definition.
(I've seen this before)
'no afternoon rest almost' is the wordplay.
'afternoon rest' becomes 'siesta' (I've seen this before).
'almost' means to remove the last letter (most of the word but not all of it).
'siesta' with its final letter taken off is 'siest'.
'no'+'siest'='NOSIEST'
'having' is the link.
(Other definitions for nosiest that I've seen before include "Most prying" , "One sits (anag)" , "Most inquisitive" , "Most snoopy" , "Most offensively curious" .)