... to anaesthetising without hesitation or any system of counting (9)
I believe the answer is:
numbering
'counting' is the definition.
('number' can be a synonym of 'count')
'to anaesthetising without hesitation' is the wordplay.
'to anaesthetising' becomes 'numbing' (I am not sure about this - if you are sure you should believe this answer much more).
'without' indicates putting letters inside ('without' can be similar in meaning to 'outside').
'hesitation' becomes 'er' (sound made while hesitating - um, er).
'numbing' going around 'er' is 'NUMBERING'.
'or any system of' is the link.
I am not very happy about this link. Some or all of it may belong to another bit of the clue.
Can you help me to learn more?
(Other definitions for numbering that I've seen before include "Enumeration" , "Counting; amounting to" .)