The point at which there is no further recourse is when the District Court makes a final order, the Court of Appeal denies the appeal, and the Supreme Court refuses to stay enforcement of the final order/appeal.
As I said above, assuming Mrs. Schiavo does not die in the interim, there are likely two rounds of appeals to go, first of the preliminary injunctive ruling and then of the final order. As it stands as I write this, the District Court denied the motion for a temporary restraining order and Mrs. Schiavo’s parents are going to appeal that denial to the Court of Appeal for the 11th Circuit in Atlanta.
If the Court of Appeal denies the appeal, the parents will almost certainly seek Supreme Court review. What would happen then is that the parents would apply for a “stay” from Justice Anthony M. Kenedy, who is the Circuit Justice for the 11th Circuit. Under the Supreme Court rules, Justice Kennedy can either rule on the application or refer it to whole court. If he rules on it himself and denies the application, it can be resubmitted to another justice. However, in the appeal of the prior federal court order in this case (in which the federal court declined to review the state court judgment prior to the special Congressional legislation this weekend), Justice Kennedy referred the application to the stay to the whole court, which denied it. (PDF of Supreme Court Order denying stay). I would presume that Justice Kennedy would likewise refer any application for a stay to the whole court. I don’t know how the Supreme Court would review on a stay application, but generally they are very rarely granted.
If the feeding tube is not reinserted after Court of Appeal and Supreme Court consideration, it appears medically likely that Mrs. Schiavo will die before additional court action, effectively ending the case. However, if the tube is ordered to be reinserted, there’s a good chance that she will remain alive for the duration of further proceedings.
What would happen in that instance runs into speculation. However, there would be a trial or hearing in District Court on the merits of the action, which would result in a final order or judgment. The losing party in the District Court has an automatic right of appeal to the Court of Appeal.
The Court of Appeal has several options on hearing an appeal. It can affirm the judgment, reverse the judgment, or affirm it in part and reverse it in part. If it reverses in whole or in part, the Court of Appeals can remand the case for further consideration in the District Court, which would allow a further appeal after the final judgment on remand. On this further appeal, the Court of Appeal would again have the same options and the possibility of remand. Theoretically, this can go on forever, but in reality, subsequent appeals usually involve very narrow points and don’t require repeated iterations.
On any ruling of the Court of Appeals, the losing party may petition for Supreme Court review. If the Supreme Court grants review, it also has the options to affirm, reverse, or affirm in part and reverse in part, and can likewise remand to the Court of Appeals or District Court. The likelyhood of multiple appeals to the Supreme Court is small, because the Court does not like to take cases unless it can make a decision that will conclusively resolve the important legal issues. However, if the Supreme Court remands, it can restart the bouncing up and down between the District Court and Court of Appeals discussed above.
That’s the theoretical path that the case can take. However, it is unlikely that this case would have many iterations as it involves narrow (though important) legal issues, but not really any factual disputes: the main question is whether there was Constitutionally required due process in the Florida state court proceedings, and the facts of what happened in the state courts are all set out in the court records and documents. Whatever the District Court decides, it is likely that the Court of Appeals and Supreme Court will have enough of a factual record to decide.
In summary, in this case, assuming that Mrs. Schiavo survives and after the question of a preliminary order reinserting the feeding tube is resolved, it is unlikely that there will be more than one trip up the federal appeals ladder.