A large policy (like $1M+) that kicks in after other policies are tapped out. Most specific underwriters (auto, boat, homeowner, aviation) won't do policies of $1M+, but if you have higher net worth you need it.

Especially as a CFI. At least you have some control over your driving habits, but if you sign somebody off you have no idea what they're going to do that might come back to haunt you.

If you're not higher net worth but are a major airline pilot, you probably still need an umbrella if you own a house, due to potential high future earnings. Normally the plaintiff will settle for the policy and leave your house alone. But if all you have is a house and future earnings, they might force you to sign over some future earnings in exchange for keeping the house (ambulance chasers prefer to deal in cash, not seized real estate).

If you don't own a pot to pee in (most CFI's) then you can just BK out and move on unscathed.