What exactly is the argument that corporations are incapable of unbounded exponential growth contra a possible future AI? Is there just something magic about computers, or am I missing something obvious?
That's key to the story of the paperclip maximizer - the paperclip maximizer will go about its task by trying to improve itself to be able to best solve the problem, and once it's improved enough it will decide that paperclips would be maximized by destroying humanity and would come up with a plan to achieve this outcome. However, humans may not realize that the AI is planning this until it's too late.
Individual corporations aren't capable of unbounded exponential growth because they can't keep the interests of the humans that make them up aligned with the "interests" of the corporation indefinitely. They develop cancer of the middle management and either die or settle into a comfortable steady-state monopoly.
Market systems as a whole can and do grow exponentially - and this makes them extremely dangerous. But they're not intelligent and so can't effectively resist when a world power decides to shorten the leash, as occasionally happens.
A human+ level AI would be able to understand and improve its own hardware and software in a way we can't with our own brains. As it improves itself, it will get compounding benefits from its own improvements.