> improving itself in an exponential growth

That seems like a possibly mythical critical point, at which a phase transition will occur that makes the AI system qualitatively different from its predecessors. Exponential to the limit of infinity.

All the mad rush of companies and astronomical investments are being made to get there first, counting on this AGI to be a winner-takes-all scenario, especially if it can be harnessed to grow the company itself. The hype is even infecting governments, for economic and national interest. And maybe somewhere a mad king dreams of world domination.

What world domination though? If such a thing ever existed for example in the US, the government would move to own and control it. No firm or individual would be allowed to acquire and exercise that level of power.