Boucher, who previously only played pickup basketball, was offered a spot on Amateur Athletic Union AAU team Alma Academy by coaches Igor Rwigema and Ibrahim Willson.

In a game versus New Jersey's Blair Academy, Boucher scored 29 points and 12 rebounds, attracting the interest of Division I college coaches.

[4] At the beginning of his first year at the University of Oregon, he worried Ducks coach Dana Altman due to his thin physique but made up for it with his energy.

In the Pac-12 tournament semifinals against California on March 10, 2017, an opposing player fell on Boucher's leg awkwardly, but he continued to play and finished the game with 10 points and four rebounds.

He ultimately went undrafted, but was signed by the Golden State Warriors to a two-way contract, the same team which purchased a draft pick from the Chicago Bulls to choose his Oregon teammate Jordan Bell.

[8] Boucher became one of the first players to sign up on the NBA's newest two-way contract policy, which went into effect that season, although his deal was not official until July 14, 2017.

On November 2, 2017, Boucher was called up to the NBA G League by the Santa Cruz Warriors and made his debut versus the Reno Bighorns.

On March 14, 2018, Boucher made his NBA debut, being called up by the Golden State Warriors to fill in for the injury-riddled lineup.

Although he was unavailable to play in the playoffs due to his two-way contract, Boucher was a part of the Warriors championship team after they swept the Cleveland Cavaliers in four games in the 2018 NBA Finals.

[23] On April 18, 2021, Boucher scored 31 points with 12 rebounds, two assists, one steal and one block and hit a career-high six threes in a 112–106 win against the Oklahoma City Thunder.

[25] With the trade of Pascal Siakam on January 17, 2024, Boucher became the final remaining player from the 2019 championship team on the Raptors roster.