Greaves is promoted by the Columbus Blue Jackets as an underdog story working himself from an undrafted goalie into the NHL, one who fans love partly because of his name and work ethic.

Greaves showed why he belongs with 21 saves for his first NHL shutout in a 7-0 win against the Washington Capitals on April 12, 2025, and 29 more to defeat the Capitals 4-1 the next night. A three-game stretch that saw Greaves stop 90 of 93 shots (.968 save percentage) earned him NHL First Star of the Week honors. On April 10 of that week, Greaves was added to the Blue Jackets roster on emergency recall and made 39 saves in a 3-2 win against the Buffalo Sabres. He finished the season 5-0-0 with a 0.80 goals-against average, .975 save percentage and his first two NHL shutouts.

The native of Cambridge, Ontario, signed his entry-level contract with the Blue Jackets on Feb. 20, 2022. He first signed with Cleveland of the American Hockey League on July 30, 2021, after making his professional debut that season for Kalamazoo of the ECHL. His 46 saves in a 4-2 loss at the Toronto Maple Leafs on April 4, 2023, set a record for a Blue Jackets debut and made him the seventh goalie with at least 46 saves in an NHL debut since 1955-56, when shots on goal began being tracked. The record at the time was 49, shared by Laurent Brossoit (for the Edmonton Oilers on April 9, 2015) and Manny Legace (for the Los Angeles Kings on Oct. 21, 1998).