To use up/hold on to something long after your share is consumed/turn is over. Alternatively, to take control away from the person whose turn it is.
"Dave had included a link to a New York Times article, and Murphy bogarted the mouse in order to click on it as quickly as possible."
-- from "Into the Fire," by Suzanne Brockmann (Ballantine Books, 2008)
To idly possess a shared item (such as a blunt) instead of letting someone else use it. Named for Humphrey Bogart, often seen in films with a cigarette idly sitting in his lips or hand.
The term 'Bogart' as it has been defined here only exemplifies part of the meaning. It has nothing to do specifically with 'hogging' a joint or smoking in general.
It is a VERB describing an act of boldness and fearlessness as was exhibited by Bogart in most of his film roles.
You Bogart your way into your boss' office when you don't care what the secretary says. You 'Bogart' your way into traffic when no one will let you into your lane and you 'Bogart' a joint when you could care less that anyone else is waiting for a hit.
This is the real origin of the term, "Don't Bogart that joint."