As much as two-thirds of the currency in circulation, according to Federal Reserve estimates. Turns out that foreigners started socking away $100 bills in large amounts after the financial crisis in late 2008. Overseas demand for the dollar had been dropping before that because the value of the greenback fell against other currencies and because international markets gained trust in their own currencies. From the WSJ:
With the higher international demand, the Fed asked the Bureau of Engraving and Printing (which executes printing orders from the central bank) to accelerate production of $100 notes during the financial crisis to meet the global needs. Most of that currency remains in circulation today, the Fed says, even though demand for $100 notes from abroad has returned to normal levels.