Day after day or day by day

'Day by day' means 'in small slow stages as each day passes', e.g. she was very ill but is getting better day by day. 'Day after day' means 'every day for a long time, often in a way that is boring or unpleasant', e.g. I’m sick of doing the same thing day after day. Neither phrase means 'one day at a time', which is a phrase used in trite advice about living your life. You could have Googled this.