Refusing to raise the statutory debt ceiling and forcing the federal government to default on its legal obligations is the ultimate nuclear weapon in congress' arsenal. Bring the president to heel by threatening to destroy the economy.
But the administration does have a counter-measure, lurking in the poorly drafted subsection (k) of 31 USC § 5112 "Denominations, Specifications, and Design of Coins":
(k) The Secretary may mint and issue platinum bullion coins and proof platinum coins in accordance with such specifications, designs, varieties, quantities, denominations, and inscriptions as the Secretary, in the Secretary’s discretion, may prescribe from time to time.
The intent here is to create collector's items, but the fact that the denominations are left up to the Secretary's discretion means that you could finance the government by minting a $1 trillion coin. It's important to be clear that both the platinum coin and the debt ceiling are purely a question of finance. The maximum amount of money the president can spend on any program is the amount congress appropriated. Conveniently, that's also the minimum amount of money the president can spend on any program. That's exactly what makes the debt ceiling so perverse. By refusing to raise it, congress denies the president a legal method of spending money that he's already legally obligated to spend. Platinum coin finance would create new spending capacity, but no new spending authority.
That said, it's mighty silly. Josh Barro suggests a reasonable compromise—pass a bill that abolishes the debt ceiling and simultaneously closes the platinum coin loophole.