An example might be something like this:
Someone spills a drink on your sweater. The person knows that she did it, but does nothing about it. She could 1) buy you a new sweater, 2) pay to have the sweater cleaned, and/or 3) apologize. Apologizing is the easiest thing to do, but the person doesn't even do that. It is the "least" difficult = easiest thing to do and she doesn't do it.
You also hear "at the very least" = the most basic thing we can expect from someone.