John Paulson won fame after he made one of the greatest financial bets of all time. What comes next? One of the largest-ever personal tax bills.

By April 17, the hedge-fund manager must make federal and state tax payments of about $1 billion, on top of roughly $500 million in taxes he paid late last year, said people close to the firm. That sum is so big it dwarfs the maximum amount the Internal Revenue Service will allow any single taxpayer to pay with a single check. (That’s $99,999,999, in case you’re wondering.)

Mr....