一番 is used before abstract nouns to mean 'a bout/spell of': 一番努力 (a great effort), 一番话 (a set of words/remarks), 一番心意 (a gesture of goodwill). 翻一番 specifically means 'to double.'
usage
As a prefix meaning 'foreign': 番茄 (tomato, literally 'foreign eggplant'), 番薯 (sweet potato, literally 'foreign yam'). These words reflect when these crops came to China from abroad.