Au bout or à bout translates to 'at the end', 'on the verge' or 'on the edge'. This is a slang expression and it's usually accompanied by de ma vie - au bout de ma vie means 'at the end of my life'.

Why do I need to know jusqu'au-boutiste? Because it's a common caricature of French politicians, which supposedly originated as a timely war reference. On ira jusqu’au bout - we will go all the way. A ...