。
She kept asking question after question.
Literal
She [topic-は] the-whole-time questions only was-doing.
~ばかりしている is the construction for 'always / only / nothing but doing X' — repetitive, often-annoying behavior. ばかり ('only / nothing but') attaches to the noun 質問 ('question'), and しているthen carries it into the ongoing-state form. Combined with ずっと ('the whole time'), the sentence captures someone in question-asking overdrive — usually with a hint of speaker exasperation or amusement. 質問する ('to ask a question') becomes 質問ばかりする when the only thing happening is question-asking.