It depends on the definition of intelligence as there are many kind/type/sort/category of intelligences and every psychologist, neuroscientist, philosopher, linguist, ethnologist, educator and a multitude of other specialist will all have their own preferred way to differentiate, categorize, regroup and make hierarchies or diagrams of all matter of intelligence and the different aspects of cognition.

Then there is general intelligence (g factor or general intelligence factor) which counterintuitively affects “intelligence” less as it increases, coined as Spearman’s law of diminishing returns (SLODR):

Tucker-Drob (2009) found that a general factor accounted for approximately 75% of the variation in seven different cognitive abilities among very low IQ adults, but only accounted for approximately 30% of the variation in the abilities among very high IQ adults.

Hence, very loosely akin to current CPUs/GPUs limits (terrible comparison, I know), there’s only so much Gigahertz we can push silicon based CPUs, there is only so many transistors we can smash together into a smaller and smaller space, there is only so much distance/area to carry tiny and fragile signals from one end of the CPU to another before it become undistinguishable from background noise, there is only so much power we can feed a tiny CPU before it reaches thermal saturation and there’s only so many cores and/or modules we can add before most of it remain dormant/barely used in day to day operations.

Now, concerning your hypothetical button, let suppose there is no such “diminishing return”, one could gladly continuously sit/walk/sleep on the button for more “intelligence”, but to keep up the brain and entire nervous system will have to drastically change just to handle all this increased intelligence. At some point even the brain volume will start to be affected and the brain would outgrow its cranium. All of it will probably excruciatingly painful and accompanied with a cocktail of neurological disorders since the brain keeps rewiring itself as it evolves.

Neat question indeed. 😆