Which Brawl mode gives the player one life to defeat all bosses?

The Brawl mode that gives the player one life to defeat all bosses is Boss Battles.

Boss Battles is a single-player endurance mode in Super Smash Bros. Brawl. Instead of fighting the playable roster in a sequence, the player faces the game's bosses, including enemies associated with the Subspace Emissary adventure. The mode tests damage management, recovery, and knowledge of each boss's attack patterns.

The one-life rule is an important distinction. Classic Mode also involves a succession of fights, but its opponents are playable characters and it uses a stock-based structure. All-Star Mode similarly has the player battle fighters, while Event Matches are individual challenges with preset conditions. Those names are therefore plausible distractors, but none describes the boss-focused gauntlet.

Brawl expanded Smash's single-player offerings substantially, most notably with The Subspace Emissary, a cinematic adventure campaign. Boss Battles sits alongside that campaign as a more direct challenge: it removes exploration and concentrates on surviving a chain of major encounters.