Will an auto insurance company pay more than the policy limits of liability under any circumstances?

Answer 1

Generally No,

The policy limits is the maximum amount of coverage afforded to the named insured for claims against them.

Remember that it is the insured client, not the insurance company, who typically is liable or who is being sued on the preponderance of some perceived liability. the insurer provides coverage for the cost of legal defense. If a judgment for a covered loss is rendered against the insured client, the company will pay those damaged on behalf of the insured up to the policy limits. They are not responsible for claimed amounts beyond the policy limits.

Sometimes Yes

Unless there is a court order specifically instructing the insurance company to pay some greater amount than the policy limits, then that's all there is.

In very rare circumstances though where an insurer was found to have exacerbated a loss through abuse, incompetence or failure to comply with some regulation, a claimant may be awarded punitive damages directly from an insurer that can exceed policy limits.