Stassi played for the Vancouver Canadians of the Class A-Short SeasonNorthwest League, and batted.280/.345/.360. In 2010, he performed well at spring training, impressing team officials, who assigned him to the Kane County Cougars of the Class AMidwest League, where he batted.229/.310/.380. He played for the Stockton Ports of the Class A-AdvancedCalifornia League in 2011, batting.231/.331/.331, but missed the end of the season with shoulder surgery. The Athletics invited him to spring training in 2012. He batted.268 with 15 home runs and 45 RBIs in 84 games with Stockton during the 2012 season.
Houston Astros
On February 4, 2013, the Athletics traded Stassi to the Houston Astros with Brad Peacock and Chris Carter for Jed Lowrie and Fernando Rodriguez. Playing for the Corpus Christi Hooks of the Class AATexas League in 2013, Stassi hit home runs in five consecutive games in July. He was named the Texas League's player of the week twice in July, and was named the Texas League's player of month for July. The Astros promoted Stassi to the majors on August 20, 2013, to serve as a backup to Jason Castro when the Astros placed Carlos Corporán on the disabled list. Making his MLB debut that day, he recorded two hits. In his second MLB game, Stassi was hit in the face by a fastball, which resulted in his being placed on the disabled list. Since the bases were loaded at the time he was plunked, he earned his first major-league RBI. Stassi began the 2014 season with the Oklahoma City RedHawks of the Class AAAPacific Coast League. He finished the season with the Houston Astros, batting.350 and driving in four runs in seven games played. Stassi played for the Fresno Grizzlies of the PCL in 2015, batting.211/.279/.384, before being promoted to the major leagues in August due to an injury to Castro. He played in 11 games for the Astros. Following the offseason trade of Hank Conger, Stassi entered spring training in 2016 expected to become the backup to Castro. In March, he had surgery on a fractured hamate bone in his left wrist. The Astros activated Stassi from the disabled list in May and optioned him to Fresno, where he batted 230/.294/.374. The Astros removed Stassi from their 40-man roster in March 2017. He played for Fresno before the Astros promoted him to the major leagues when Brian McCann was injured on August 14. Stassi went on the disabled list on August 25 with left hand inflammation caused by a blister. In 14 games in 2017, Stassi finished with 2 home runs and 4 RBIs and batted.167/.323/.458. The Astros finished the season 101-61, and eventually won the 2017 World Series, their first ever championship. Stassi did not see any postseason action, but was still on the 40-man roster at the time and won his first championship title. Stassi was named to the Astros' Opening Day roster in 2018 as a backup to McCann. In 221 at bats in 2018 he batted.226/.316/.394. On June 26, Stassi came into the game as pitcher in the 9th inning in a blow out loss to the Pittsburg Pirates. in In 2019, Stassi had a limited role with the Astros as a backup catcher to Robinson Chirinos and only appeared in 31 games where he had just 15 hits in 90 at-bats with one homerun and three RBIs.
Los Angeles Angels
On July 31, 2019, the Astros traded Stassi to the Los Angeles Angels in exchange for minor leaguers Rainier Rivas and Raider Uceta.
Personal life
Stassi's father, grandfather, and great-grandfather played as catchers in minor league baseball. His father, Jim, coached him at Yuba City. Max's great uncle, Myril Hoag, played in three World Series, batting.320 in them. His brother, Brock Stassi, has also played in MLB. Max and Brock Stassi have raised more than $100,000 with an annual charity event held in Yuba City, called "Homers for the Hungry." The money goes to local food banks.