You can get exponential growth from completely ordinary feedback loops. You start with some amount of stuff, you do a series of steps and you end up with more of the same stuff you started with. As you keep going through the loop, the stuff you have grows exponentially. That's for example how exponential economic growth works.
Of course data, compute and model size are not held constant. You start with some money and use it to acquire researchers, data and compute, and have the researchers produce a big model and you use that model to get more money, and you use the additional money for more researchers, more data, and more compute to produce a bigger model. This is what has propelled exponential AI progress so far.
Recursive self-improvement is invoked to predict superexponential growth. The idea is that instead of only using the model to make more money, you add it to the researchers to speed up the loop, so not only is the money growing with every iteration, the iteration time also gets shorter, producing growth that is faster than exponential.
The problem with this simplistic prediction is that it assumes additive and multiplicative relationships of the form money = (researchers + AI)×compute_spend, but if doing more research paid off so reliably, you could also just hire more researchers, abstractly money = research_spend×compute_spend and with a balanced allocation of research and compute, you would get a money-squaring machine even without using AI for AI research.
And the reason this doesn't work in reality is that there are diminishing returns everywhere. You can also see this in the OpenAI post, where they write 7 times as much code to run 1.6 times as many experiments, and those additional experiments probably only result in minor improvements to model quality.