Zoom out fractals?

It is well known that if we zoom in on the Mandelbrot set we get selfsimilarity. So I wonder if $g$ is a fractal (in the complex plane) generated by a nonperiodic nonpolynomial entire function $f$

$g:= f(f(...))$

Is it necessary or even possible that the fractal is infinite in size and that when we zoom out , we get selfsimilarity too ?

Does the existance of zoom out fractals require that the fractal is also a zoom in fractal ?

What is the formal way or term to express ' zoom out selfsimilarity ' or ' zoom out fractal ' , if any ?

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How do you mean this $g=f(f(\dots))$? –  Berci Sep 30 '12 at 14:49
@Berci : Well let $f$ be $z^2 + 1$ , then you get the Mandelbrot. I think that example is clear. –  mick Sep 30 '12 at 16:53