Papagiannis began his basketball journey in the youth system of Peristeri in his native Greece, and made his professional debut with the same team during the 2011-12 season. In the 2013-14 season, he played high school basketball in the United States at Westtown School. The following year, he returned to Greece and joined Panathinaikos for the 2014-15 season. After two seasons with Panathinaikos, Papagiannis was selected 13th overall in the first round of the 2016 NBA Draft by the Phoenix Suns, making him the highest-drafted Greek player in history. His rights were then traded to the Sacramento Kings, where he spent two seasons before joining the Portland Trail Blazers in 2018. That summer, he returned to Greece to rejoin Panathinaikos, where he played for five seasons. In the 2023-24 season, Papagiannis played in Turkey for Fenerbahçe Beko, before signing with AS Monaco ahead of the 2024-25 season. During his time with Panathinaikos, Papagiannis won three Greek League championships and four Greek Cups. In the 2023-24 season, he won the Turkish Basketball Super League title and the Turkish Cup with Fenerbahçe Beko. In the 2021-22 EuroLeague season, Papagiannis led the league in both rebounds (8.2) and blocks (1.7) per game, earning a spot on the All-EuroLeague Second Team. In the 2024-25 season, he once again topped the EuroLeague in blocks, averaging 1.3 per game. He was named to the Greek League Best Five in both the 2020-21 and 2021-22 seasons and led the league in blocks during the 2021-22 campaign. In the 2024-25 season, he also led the French League in blocks, averaging 1.4 per game. While playing for AS Monaco in the EuroLeague this season, Papagiannis appeared in 29 games, averaging 16:30 minutes, 7.3 points, 3.4 rebounds, and 0.5 assists per game. In the French League, he played in 32 games, averaging 17:50 minutes, 5.3 points, 3.4 rebounds, and 0.6 assists per game.