Final answer:
In REBT, 'shame-attacking exercises' is the technique where clients do what they normally avoid because of societal judgment to overcome their fear of embarrassment.
Step-by-step explanation:
The technique in Rational Emotive Behavior Therapy (REBT) that involves having the client do the very thing they avoid because of 'what people might think' is known as shame-attacking exercises. This technique is designed to help individuals confront and overcome their fear of public embarrassment or judgment. By intentionally engaging in behaviors that may elicit disapproval or ridicule in a controlled setting, clients learn to diminish their concerns about others' opinions and strengthen their self-acceptance.