In the early 1990s, Denis Leary hit the comedy circuit with a bang with his No Cure for Cancer routine, which touched on themes around smoking, drug use and political correctness.

Leary might be tempted to jump aboard a bandwagon that finds today’s comedians to be less brave and risky than when he first came up, but he eschews that temptation.

“Comedy is not everybody’s cup of tea but god damn you think you’ve seen everything and then you see someone else coming who makes you laugh your ass off,” he told Deadline. “Generationally, there are always [new] people.”

Leary points to the way in which “edgy” comedy greats one generation below him like Bill Burr and Dave Chappelle smashed barriers, but he is also enthused by younger comedians such as Daily Show performer Ronny Chieng. “He came over to America at a young age and cut his...