Sovereign governments such as the United States can print new money. However, there's a statutory limit to the amount of paper currency that can be in circulation at any one time.
Ironically, there's no similar limit on the amount of coinage. A little-known statute gives the secretary of the Treasury the authority to issue platinum coins in any denomination. So some commentators have suggested that the Treasury create two $1 trillion coins, deposit them in its account in the Federal Reserve and write checks on the proceeds.
The government can also raise money through sales: For example, it could sell the Federal Reserve an option to purchase government property for $2 trillion. The Fed would then credit the proceeds to the government's checking account. Once Congress lifts the debt ceiling, the president could buy back the option for a dollar, or the option could simply expire in 90 days. And there are probably other ways that the Fed could achieve a similar result, by analogy to its actions during the 2008 financial crisis, when it made huge loans and purchases to bail out the financial sector.
The "jumbo coin" and "exploding option" strategies work because modern central banks don't have to print bills or float debt to create new money; they just add money to their customers' checking accounts.