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Hi, I'm new to these forums and so I haven't managed to get my head around ultrafractal yet, but I've got a couple of (horribly rough) pictures of a 3d fractal idea.
It works just like the mandelbulb but instead of polar (longitude/latitude) parameters, it operates on a torus, meaning no singularity points on the shape.
I've transformed the torus back into a square patch (notice how it tiles).
Also notice that the front wall traces out a 2d mandelbrot and the side walls trace out a similar fractal shape.
There are some free parameters, the ring radius and the chosen scale on C.. the two are set to 2 and 1/4 respectively, and it exits when distance from torus > 2.
(http://fractal1.jpg)
(http://fractal2.jpg)

The formula is like so (c++):
    ringWidth = 2.f;
    cScale = 0.25f;
    power = 2.f;
    phi = fmodf(phi*power, 2.f*PI); // rotate around ring girth
    theta = fmodf(theta*power, 2.f*PI); // rotate around big ring circle
    radius = powf(radius , power);  // scale distance away from ring
    pos = Vector((ringWidth - radius *cosf(phi))*cosf(theta), (ringWidth - radius *cosf(phi))*sinf(theta), radius *sinf(phi));
    if (radius  > ringWidth)
      return false;
    pos += C*cScale;
    theta = atan2f(pos.y, pos.x);
    float dist = ringWidth - sqrtf(sqr(pos.x) + sqr(pos.y));
    phi = atan2f(pos.z, dist);
    radius  = sqrtf(sqr(pos.z) + sqr(dist));

OK, I got ultrafractal working and I think I've convinced myself that this line of reasoning is a no goer.
The idea is to keep singularities in the rotation part (which occur at the north and south poles on the mandelbulb) outside of the fractal by operating on a donut.
But ultrafractal shows that any variation on this idea will inevitably produce a fractal of donuts (lots of rings with hole in them) which isn't what we want.. and it doesn't actually seem to remove the whipped cream effect as I hoped.
The above pictures just aren't detailed enough to show the hoops.
Was fun trying though  :D

though, it's a nice idea :)
Maybe you can do a new 2D-Variation with it? (you could flatten the results out as you showed here and then tile them... could give interesting pictures :) )

Heh, I tried a torus coord approach too a while back without success. Those pics actually look a little bit promising, but I'll take your word for it if you had no luck in UF.