Depends on how long you let it sit.
Caffeine is water soluble, the esters and oils that give coffee its flavor are heat-released. The longer you let the water and grounds sit, the more caffeine will eventually be extracted. That's why hitting fine grounds with rapid steam results in an extremely concentrated, but not highly caffeinated, brew of espresso.
The first cup of french press isn't that bad, caffeine-wise, but if you let it sit for a while and then come back for a cup later, it can jazz you right up.