Maybe one day I'll make a version that works on a web page but for now it is only available as a Windows program. Surprisingly the answer is 0. Each set of triangles that is removed reduces the remaining area by a quarter, and so eventually nothing is left. This fern really is a Spirofractal image. The realistic colouring arises because fractals often look like objects in the natural world, and one of the colour schemes in Spirofractal mainly uses natural looking colours.
Many Spirofractal images look like some strange animal or plant.
Some resemble something you might see down a microscope. This impression can be enhanced by using one of the nice features of Spirofractal. You can double click on any part of the image, and the fractal will be drawn again with that part in close up. You can do this as often as you like. Unfortunately this inevitably causes it to take much longer to draw the image because the many points that are no longer displayed still must be calculated.