3. Basilica of the National Shrine of Our Lady Aparecida
According to its website, this is the largest shrine in the world dedicated to Mary, Jesus' mother — and there's a bit of a story behind it. In 1717, three fishermen had been ordered to fish in the River Paraiba (which runs beside the Basilica) to provide for a banquet honoring a visiting governor. After a while, they still hadn't caught a single fish when one of the men decided to throw his net in for the last time. He pulled out a statue of the Virgin Mary. One of the fishermen kept the statue in his house until a church was built to accommodate the number of pilgrims coming to pray to her. (Side note: the men also had tremendous luck catching enough fish for the banquet.)
The statue of Our Lady of Aparecida is currently held in the National Shrine, where some 11 million pilgrims visit each year, including three popes. Most recently, Pope Francis came to the basilica during his stay in Brazil for World Youth Day.