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I know the liver is responsible for doing lots of stuff to the blood, including both adding glucose and removing "weird stuff" (stuff destroyed by hepatocytes). And I know that statistically every volume element of blood will eventually pass through the liver.
However, if you were magically minaturized a la Fantastic Voyage, and traveled with the blood flow, making ideal turning decisions at every arterial branching but never going upstream or out of the blood vessels themselves, could you completely avoid 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.