Well, it all depends on what you consider the “brain”. There are three parts of the brain, two of which is technically optional.
The Cerebrum, the big wrinkly part that makes up most of the brain, is where you do all of your thinking, sensory processing, speaking, voluntary movement, etc. Everything that makes you “you” resides there. That said, you can and people do live with that part of the brain completely inactive or absent. They are total vegetables, but they can live.
The Cerebellum sits below the cerebrum. I believe that all of its functions are still a bit mysterious, but it processes voluntary movement, handles balance, perception of time, etc. Again, you can live without it, but you wouldn’t want to. Even if your cerebrum was intact, your quality of life would be abysmal.
And finally, the Medulla Oblongotta is almost completely encased in the Cerebellum. You CANNOT LIVE without this part of the brain. It handles all automatic functions (heart beat, breathing, etc.) and connects the spinal cord to the rest of the brain. Hydrocephalics and Anacephalics may be missing their cerbrums and even their cerebellums and clincally live, but if they don’t have the medulla oblongotta, they’re dead.
Of course, with modern life support they can keep people alive without a head, but I don’t think that’s what you had in mind.