adverb, adverb taking the 'to' particle
Describes a slow, steady change or progress happening each day. Often used with verbs of change like 変わる, 増える, 成長する.
As spring approaches, it's getting warmer day by day.
Refers to each individual day as a unit, often in contexts of daily effort, routine, or counting days. More concrete than the adverbial sense.
彼の病状は一日一日と回復に向かっている。
His condition is improving gradually, day by day.
I want to live each day to the fullest.
Every day is precious until the exam.
Reduplication of 一日 (いちにち, 'one day'), emphasizing each day individually. The adverbial sense with と developed from the idea of 'day by day' leading to gradual change.