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So fractals are something which fascinate me a lot. Most of the fractals I've seen till date are mostly formed by triangles or, precisely, straight lines. can fractals be formed by using something like, circles or spirals?

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    Do you consider the Mandelbrot and Julia sets to be composed of straight lines? – 2017-01-27
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    In my opinion, yes, because ultimately when we zoom them up, they are composed of straight lines – 2017-01-27
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    "ultimately when we zoom them up, they are composed of straight lines" - doubtful; being able to do that "implies" (in a way that could be made more rigorous) differentiability of that perimeter. – 2017-01-27
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    well, I don't have much knowledge, just started studying about them ...I am sure its a long way to go – 2017-01-27

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As observed by others, fractals can be built from lots of things. A good example is the Apollonian gasket; examples with less intuitive constructions include the Julia sets, their big cousin the Mandelbrot set, Newton fractals, and fractal flames, among many, many others. There's also the Wada basin fractal which isn't so geometric in nature, but its "construction" only involves spheres.

Another, somewhat sillier example is Romanesco broccoli, an excellent example of a natural fractal (though I'm not enough of a botanist to know whether it's somehow composed of straight lines).

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Fractals can be composed of pretty much anything. You of course have the Koch Snowflake and Hilbert Curve, which are cool, but you should look into the following:

  • Sierpinski Curve and Menger Sponge, fractals from a plane and a cube

  • Mandelbrot/Julia sets, fractals formed in the complex plane from recursion

  • Weierstrass function, which shows fractal properties, and while it's still "a line", it's certainly interesting and strange enough to differentiate (no pun intended) itself from the other "line-made fractals"

A bunch of methods for making fractals can actually be found here, I'll bet some of those are the kinds of things you're looking for.

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The fractal objects in this paper by Vincent Borelli et al. are made exclusively of curves. http://www.pnas.org/content/109/19/7218.full

In fact, the method in the paper is very general, and used to create $C^1$ isometric embeddings of surfaces in three dimensional space when the curvature of the surface to be embedded prevents it.