to get as much money or as many advantages as you can from a situation, in a very determined and sometimes dishonest way
milk somebody/something for something
Their landlord regularly milks them for extra money by claiming for damage to his property.
He seems to be milking the incident for all it's worth (=getting as much from it as possible).
to take milk from a cow or goat:
I helped to milk the cows.
—milking noun [uncountable]
They had risen at 5.30 to do the milking.