I assume re-insurers use Lloyd's of London to split/sell very large risks. This is similar to syndicate loans, where a group of banks agree to split a single very large loan for huge construction projects. Although, these days, syndicate loans are much less popular than directly issuing debt via capital markets (if possible).

Usually there are a few deep players Berkshire Hathaway(who happens to own a reinsurance business or two) has about $150 billion in cash just laying around, so they can pretty much insure anything they want and backstop it themselves.

But for those without such a deep bench, they, like other commenters mentioned do a round robin sort of thing.

Though you don't have to have 150 billion USD in cash. You can also have, say, 250 billion USD in the stock market instead.

(I went with a bit more, because stocks are more volatile than cash.)

It can be circular rather than nested. You may reinsure a peer company’s auto policies while they reinsure your home policies.

But that's not really circular, is it?

If they have to pay out on their reinsurance of your home policies the fact that you've reinsured their auto policies does not help them.

Someone has to insure their reinsurance of your home policies, don't they?

Insurance coverage is typically limited. So even if you caused a billion dollar in damages, your insurance would only be on the hook for, say, a million dollar.

So there's a limit to how much they need to re-insure.

Someone already said that the rabbit hole ends with eg Berkshire Hathaway. But it also ends with eg (retail) investors who hold catastrophe bonds or shares of insurance companies.

Catastrophe bonds, which impose losses on holders in the event of a catastrophe, are an increasingly popular way to re-insure or re-re-insure. Because they can be widely held and each holder can hold them as part of a diversified portfolio, the risk can be dispersed much more widely.

You've described part of the 2008 financial crisis.

Pretty much what I was thinking. A number of seemingly uncorrelated risk that, scratching the surface, turn out to be highly correlated.