There is a long history of presidentially-appointed people saying they "serve at the pleasure of the President." It simply means that they hold the job only so long as the president wants them there.

This has extended "serving" to broader, more casual usage when speaking about people who work for the president as you can see here. It's not "serving" like working as a servant/a billionaire doing whatever the president says in a controlling/exploitative way, it just means that Larry King thinks that Ted Turner wouldn't turn down a cabinet appointment.

You can also infer from the way King says that "what's he got left in life to gain," and Clinton's surprised "you're kidding," that they both consider this kind of service to be a major downgrade from the billionaire life - King saying here that Ted Turner has nothing left to gain indicates a separation in his mind between personal gain and working for the Clinton administration. Clinton's surprised response tells us he feels the same way, surprise that a billionaire would want to work for the president.