It seems like most people expect the emergence of the strong AI to be a sudden event, something similar to the creation of a nuclear bomb. However, it's far more likely that AI will undergo gradual development, becoming more and more capable, until it is similar in its cognitive and problem solving abilities to a human. It's likely that we won't even notice that exact point.

I'm not even sure governments are interested in developing AGI. They probably want good expert systems as advisers, and effective weapons for the military. None of those require true human level intelligence. Human rulers will want to stay in control. Building something that can take this control from them is not in their interests. There likely to be an arms race between world superpowers, but it will probably be limited to multiple narrow AI projects.

Of course, improving narrow AI can lead to AGI, but this won't be the goal, IMO. And it's not a certainty. You can probably build a computer that analyses current events, and predicts future ones really well, so the President can use its help to make decisions. It does not mean that this computer will become AGI. It does not mean it will become "self-aware". It does not need to have a personality to perform its intended function, so why would it develop one?

Finally, most people think that AGI, when it appears, will quickly become smarter than humans. This is not at all obvious, or even likely. We, humans, possess AGI, and we don't know how to make ourselves smarter. Even if we could change our brains instantaneously, we wouldn't know what to change! Such knowledge requires a lot of experiments, and those take time. So, sure, self-improvement is possible, but it won't be quick.