Nameless dog at home — one who kept escaping (7)
I believe the answer is:
houdini
'one who kept escaping' is the definition.
(Harry Houdini was an escapologist)
'nameless dog at home' is the wordplay.
I cannot really see how this works, but
'at' could be 'in' and 'in' is found within the answer.
'home' could be 'ho' (abbreviation for house) and 'ho' is present in the answer.
The remaining letters 'udi' is a valid word which might be clued in a way I don't understand.
This explanation may well be incorrect...
(Other definitions for houdini that I've seen before include "Harry . . . . . . ., famed American escapologist" , "being given to escapism?" , "Harry was famous as an escape artist" , "one who'd try to escape?" , "Showman" .)