Is there a closed path through the circulatory system that never passes through the liver?
The liver gets blood from two sources: the hepatic artery and the portal system of venous drainage from the gastrointestinal tract.
As long as you don't follow one of these paths, you'd avoid the liver. Some examples would be circulating through any of the limbs or head, but any path that avoids those two paths will do. As you point out this is statistically unlikely without your magical navigation abilities.