I seem to recall back-to-back doubleheaders one year (?)...perhaps a mindfact.
In 2018, the Nats played a doubleheader against the Cubs and then their next two games were a doubleheader against the Phillies, but they weren't on consecutive days. The Cubs games were on September 8 and the Phillies games were on September 11. The intervening days would have seen a game against the Cubs on September 9 and a game against the Phillies on September 10, but both were rained out. (The Cubs games were at home and the Phillies games were on the road, too.) The Nats swept both doubleheaders, incidentally.