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Besides the Cantor dust, i can't think of any.... 

• What do you count as 1D
• What do you count as interesting?

Technically, the simplest cellular automata are 1D. It's just the typical visualization, where you put time dimension (e.g. iteration count) in a second spacial dimension so you can see how the interesting patterns unfold.

The same thing is true for how Cantor dust typically is depicted, as well as for the Logistic map.

Just as well, however, you could think of the generic cellular automaton as a high-dimensional fractal of which each individually valued automaton (e.g. rules 30 or 110) are mere 1+1D slices of that.

Plotting the Mandelbrot set's real axis as a typical f(x) plot reveals some rarely explored patterns as well. They might not be all that interesting but they are there.

If you fold a long piece of paper alway in the same direction and then unfold it all the way, you get a fractal pattern of valley and, uh, hilltops? - if you map those to 0 and 1, what you have is essentially a discrete 1D fractal pattern. If you interpret those 0s and 1s as left and right turns of 90°, you'll end up with the 2D dragon-curve which, however, is only a line.

Similarly, all space-filling curves at least on some level can be seen as 1D objects. Of course, those monster-curves were traditionally what first really challenged simple, discrete dimensionalities, so their case isn't so simple.

That being said, if you strictly only allow 1D-representations where you can't clarify what's going on by a "fake" (purely representational) second dimension, only so much can be done and most of it might not be all that interesting to look at.

Many things you'll try to keep in 2 or 3 dimensions, simply because those are the cases we can deal with most easily and efficiently.
1D structures tend to be way too wide and thin for our eyes to be read easily.
With 3D, things already start to obscure and only because we learned to live in a 3D world can we make a good amount of sense from such representations.
If you really shoot for representational clarity, you usually want to go 2D. It's as easily overviewable to us as 1D while putting information into a much more compact form without yet occluding anything like 3D would.

see, that's why I asked

don't know if this matches your definition of interresting, but although one might not await, it is a 1d fractal curve. Menger himself prooved that!

Absolutely!
just for the heads up:
http://www.fractalforums.com/ifs-iterated-function-systems/dragon-curve-featured-in-jurassic-park-%28book%29-explained-by-numberphile/

   Contemplate this:

  .

How about a number line?

0 1 2 3 4 5 6 7 8 9 10 ...
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2 ...

The idea is:

The number has got x digits after its point/comma.
The marked numbers follow this rule:
after removing the point/comma, it is a product of x+1 prime numbers.

The only problem is, it might not have a repeating visual pattern.

What about p^n for any real p > 0, p != 1, n integer...   that doesn't have the awkward problem of imperfection...

However these are only "self-similar in one direction", unlike say the Cantor dust which is "self-similar in two directions", so I think it's less interesting.