。
She drove my car without so much as asking.
Literal
She [topic-は] any notice [even-も] without-に I [possessive-の] car [object-を] drove.
断り is one of those Japanese words whose surface meaning surprises learners — it can mean 'refusal' but here means 'notice' or 'permission asked,' i.e. checking with someone before doing. 何の断りもなしに literally 'without any notice at all' is a small fixed cluster: 何の + N + も + なしに carries the 'not even a single' force. ~なしに sounds slightly more literary than ~ないで; in casual speech 黙って ('silently / without saying anything') would be more typical for the same idea.