Who is the most important stakeholder outside the finance function that finance leaders/the CFO need to have a strong partnership with?

My view is that the business team running customer contracts for the delivery of products or services are the most important stakeholder outside Finance. We need to provide them the BU and Projects profitability and have conversation about profitability improvements. We also need to work closely with them to take business and accounting actions based on the profitability review. 

I would say Business head or leader / ceo in general except if there is a business transformation underway I would add chro and cio as additional stakeholders

Depends on industry. I have seen usually with COO and/or Business Unit heads.

My view, this is the CEO of the business. He dictates the strategy and steers the engagement of the operating board of the business. All of who, finance need to business partner, provide service to, and generate transformation for finance, through digitization and process alignment.