Upside-down on horse, exciting at first in infancy (6)
I believe the answer is:
nonage
'infancy' is the definition.
(I've seen this before)
'upside-down on horse exciting at first' is the wordplay.
'upside down' says the letters should be written in reverse.
'horse' becomes 'nag' (nag is a kind of horse**).
'at first' indicates taking the first letters.
The first letter of 'exciting' is 'e'.
'on' in reverse letter order is 'no'.
'no'+'nag'+'e'='NONAGE'
'in' acts as a link.
(Other definitions for nonage that I've seen before include "In law, state of being under age for certain things" , "very early life" , "Period of youth before maturity" , "State of being unable to marry" , "Period of immaturity" .)