Jon Ambrose was lying in bed playing a word game with his wife on her iPhone last winter when she left the room for a few minutes. After 12 years of marriage, the 33-year-old computer engineer got his first close look at her phone’s home screen.

What he saw shocked him: thousands of unread emails, 45 pending app store updates, a dozen-plus unchecked voice mails. The phone was “overflowing with ridiculousness,” says Mr. Ambrose, a religious pruner of both his inbox and his phone’s home screen.