Inclination of number extremely cagey about lair (8)
I believe the answer is:
tendency
'inclination' is the definition.
(I know that inclination can be written as tendency)
'number extremely cagey about lair' is the wordplay.
'number' becomes 'ten' (ten is a kind of whole number).
'extremely' means to remove the middle letters (only the extremes of the word are used).
'about' says to put letters next to each other.
'lair' becomes 'den' (synonyms).
'cagey' with its centre removed is 'cy'.
'cy' after 'den' is 'dency'.
'ten'+'dency'='TENDENCY'
'of' acts as a link.
(Other definitions for tendency that I've seen before include "Likelihood - inclination" , "Inclination or leaning" , "Proclivity" , "Propensity, intensity" , "Predisposition" .)