Are Fractals sequences Finite or Infinite?

Are self-similar fractal sequences to be considered finite or infinite? Qualitatively they seem finite after all they are self-similar but quantitatively they go to infinity...
Framed in a different way are there infinitely distinct patterns if one considers space is quantized instead of continuous? Even if a quantized space is infinite in size should we not at some point start seeing a repeat of the patterns at different scales just by random chance? The point is is infinity just a boundless illusion or is there real qualitative infinite diversity of shapes? Why do fractals tend to stumble on these universal attractors?

...if we consider that there is always more quantized space then quantized matter in an infinite set then patterns have a bigger then zero chance of repeating thus rendering infinity a flawed concept. On the other hand, one can argue that one infinity is just bigger than the other hence why patterns repeat...

...the fact remains that no matter how much you zoom in in a fractal sequence the amount of information in it is finite and that is the point regarding the problem with infinities. To me, they look delusional and irrational. Counting endlessly with a finite information set does not make much of infinity...