If you're unbeholden, you don't owe anyone anything — you don't need to feel an obligation to say "Thanks."
This formal adjective takes beholden, "owing thanks in return for help," and adds the prefix un- ("not"). If your friend helps you out by walking your dog every day while you rest your sprained ankle, you are beholden to her — obliged to say thanks, buy her a gift, or offer to walk her dog in return. But if she refuses to walk your dog, you're unbeholden to her. She wasn't helpful, so you aren't obligated to thank her!