Actually, the answer is probably "it's complicated". https://en.wikipedia.org/wiki/Force_between_magnets#Force_be... gives an approximation in the x >> R case, but in the pogo stick situation we're not in that case, especially once we get to 5 magnets or so. But even there, if x >> L also holds (which it would in this case, since the magnets shown have R and L of about the same size), the force goes as 1/x^3 or so, because the 1/x^2 terms cancel out if you go through the small-L approximation. Which, again, is not relevant to the problem because our case has small x.

For small separations, the exact geometry of the magnets starts to matter. http://www.magnetsales.com/design/faqs_frames/FAQs_2.htm has some experimental results that show the field going up a bit over 3x when distance drops from 1 to 0.5, then ~2x when going to 0.25, then ~1.5x when going to 0.125 and only changingby 1/8 when going to 0.063... all distances in inches from _surface_ for a magnet that's radius 0.5 and length 0.5 in the same units. Note that this means that the distances from the _center_ are approaching 0.5.

All that said, the force is almost certainly not exponential, I would guess.