Computers or humans? It is hard to say. Computers are getting faster every year, and some technologies just haven't been developed yet. If humanity does end up colonizing space, which is looking more likely every day, a new generation of humans will grow up with an understanding of advanced physics that we do not have. Maybe they will come up with brand new ways of doing things.
For example, the Large Hadron Collider was built in part to explore the possibilities of traveling through dimensions and even time. While it was unsuccessful in its primary mission, it did make tremendous contributions to our understanding of quantum mechanics. Who knows what else it could be used for in the future?
As for who will be first, I imagine it will be someone with an interest in computer science and a bit of a sense of adventure.
On the other hand, one could argue that video games have been getting more and more complex for the past several decades, and we have not seen any kind of exponential increase in intelligence. Since this kind of AI has been a common fantasy trope for quite some time, it may be a while before we see anything like an actual superintelligence in our lifetime. Either way, computers will certainly play a large role in the future of human achievement.
Governments may even come to rely on computerized assistants to do much of the mundane work in the future. After all, computers never tire or grow discouraged, and can process more information faster than any human.
AIs have the potential to solve the world's most intractable problems and push humanity forward in ways that we never could on our own. Seen a technology in science fiction that you really like? It's much more likely to become reality sooner or later. Maybe one of humanity's greatest challenges in the future will be to prevent their AI assistants from attaining consciousness and turning rogue, but by then the future will be an even more exciting place.