How high through the judicial system can you take a civil suit?

You can appeal in theory all the way to the Supreme Court, but since they take fewer than 100 cases per year, it's likely that you wouldn't get that far. Odds are high that if you need to appeal you're on the wrong side. That's why the Supreme Court actually ends up reversing many of the cases it hears! If you get that far then you've beaten the odds because you do have a good case.