elastic potential energy? yes. Elastic kinetic energy? Yes. elastic "non-potential" energy? nope, but even if I had I'm pretty sure it doesn't work the way you were saying. besides which, the surface area wouldn't be infinite, as Fictionpuss said it would asymptote around a certain value, probably just less than the surface area of the largest rendition of the fractal. Say the largest rendition had an area of 1 m^2 it most likely wouldn't ever hit 2m^2 total. This is because, while there are infinite renditions, each rendition is smaller than the last to the point where the additions become insignificant, each a fraction of the last. Say we take a common recursive pattern, the Sierpiński carpet. This carpet is begun with a square split into nine parts with the middle part removed. The smaller squares are then divided into nine parts with the middle part removed, and so on, giving us something like this;
#########
# ## ## #
#########
### ###
# # # #
### ###
#########
# ## ## #
#########
In this case, surface is removed rather than added, but the principle is the same. Suppose the surface area was initially 81m^2. That makes each # worth a m^2
after the first removal it would remove 9m^2 giving us 72m^2.
after the second it would remove 8m^2 giving us 64m^2
and it would continue to get smaller and smaller, but each time it would remove a smaller portion. Eventually, it would approach 0, but it would never reach 0. It is the same with adding surface area except you would never double the surface area rather than reduce it to zero.
Edit: fixed some basic math failure.