What does "exhausted" mean in this sentense?
In the case of a judicial challenge to the tax liability, teh liability is not finally determined until all judicial appeal rights have been exhausted.

exhausted = consumed, used up, completed.  Here there may be several courts, like Supreme Court, and each case may have a right of appeal.  Suppose there are three available appeals.  Once the first second and third appeal have been denied, the second rights of appeal have been exhausted.