Yes, I left out some code for the sake of brevity. Try this:

ParametricPlot[Sum[{1/(Sqrt[2])^k Sin[(Sqrt[2])^k t], 
   1/(Sqrt[2])^k Cos[(Sqrt[2])^k t]}, {k, 0, 15}], {t, 0, 100 Pi}, MaxRecursion -> 10]

Reduced values of t and k, and a high MaxRecursion, should help. Give it about 30-60 seconds to evaluate. With this, you can begin to see the image take shape. However, you will need k > 20 and t > 500Pi to see the details fully emerge, and PlotStyle-> Thickness has to be just right, otherwise it will appear too dark or too faint. I also assembled the plot piecewise, taking t just 10 Pi at a time, because it is very computationally intensive due to high values for t, k, and MaxRecursion. Here is the actual code I used:

CoolCurve[i_] := ParametricPlot[
   Sum[{1/(Sqrt[2])^k Sin[(Sqrt[2])^k t], 1/(Sqrt[2])^k Cos[(Sqrt[2])^k t]}, {k, 0, 20}]
   , {t, (10 i - 10) Pi, 10 i Pi}, PlotStyle -> Thickness[0.0003], MaxRecursion -> 10, ImageSize -> Large];

Export["fhqwhghads.jpg", Show[Table[CoolCurve[n], {n, 0, 120}]]]

It will take at least a couple hours. The image looks a little saturated (too dark)... If I were doing it over again, I would use Thickness[0.0002], and ImageSize->{5000,Automatic} to make a larger, more detailed image. I chose jpg format because .tiff would have been a gigabyte or more. Make sure you have your jpg compression settings to 0.00 to avoid distortion. You can check this by right-clicking any plot, Save Graphic As, choose JPG, click Options, set compression to 0.

Also, if you want to indulge in a little pareidolia for fun, you can see what look like faces along the vertical axis :) . Stay tuned, two more plots coming.