As you've observed, when you're dealing with that amount of money, you're going to have to give up FDIC guarantees. That means that keeping the money in a bank account carries some risk with it: if that particular bank goes bust, you could lose most of your money. There are a few options to stretch the FDIC limit such as CDARS, but likely can't handle your hypothetical $800 million. So, what's a lucky winner to do? There are a few options, including treasury securities, money market funds, and more general capital investments such as stocks and bonds. Which one(s) are best depend on what your goals are, and what kind of risks you find acceptable.
Money in the bank has two defining characteristics: its value is very stable, and it is liquid (meaning you can spend it very easily, whenever you want, without incurring costs). Treasury securities and money market funds each focus on one of these characteristics.
A treasury security is a piece of paper (or really, an electronic record) saying that the US Federal Government owes you money and when they will pay it back. They are very secure in that the government has never missed a payment, and will move heaven and earth to make sure they won't miss one in the future (even taking into account recent political history). You can buy and sell them on an open market, either through a broker or directly on the Treasury's website. The major downside of these compared to a bank account is that they're not as liquid as cash: you own specific amounts of specific kinds of securities, not just some number of dollars in an account. The government will pay you guaranteed cash on specified dates; if you need cash on different dates, you will need to sell the securities in the open market and the price will be subject to market fluctuations.
The other "cash-like" option is money market funds. These are a type of mutual fund offered by financial companies. These funds take your money and spread it out over a wide variety of very low risk, very short term investments, with the goal of ensuring that the full value will never go down and is available at any time. They are very liquid: you can typically transfer cash quickly and easily to a normal bank account, write checks directly, and sometimes even use "online bill pay"-like features. They have a very good track record for stability, too, but no one is guaranteeing them against something going terribly wrong. They are lower risk than a (non-FDIC-insured) bank account, since the investments are spread out across many institutions.
Beyond those two somewhat "cash-like" options, there are of course other, more general investments such as stocks, bonds, and real estate. These other options trade away some degree of stability, liquidity, or both, in exchange for better expected returns.