The Supreme Court, eventually.
There are three paths the case could take on its way there, according to Josh Blackman, a law professor at South Texas College of Law. First, the Trump administration could file an emergency application with the Supreme Court, once again seeking a stay of the restraining order. That process could move quite quickly. The New York Times reported that that was the most likely course of action, and that a decision as soon as next week is possible.
Second, the White House could file an emergency petition and also petition for certiorari — the slower, more traditional path to the Supreme Court. This has an added benefit for the Trump administration: By the time the case was heard by the court, Trump’s nominee Neil Gorsuch could have a seat on the bench.
Third, they could bide their time with respect to the Supreme Court. They could simply let the case play out in the lower district court in Seattle — the judge there still has to decide on issues beyond the temporary restraining order. Or they could petition the full 9th Circuit en banc — 11 judges as opposed to just the three-judge panel.