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Just elaborating on the question:

We all use to natural numbers as dimensions: 1 stands for a length, 2 for area, 3 for volume and so on. Hausdorff–Besicovitch extends dimensions to any positive real number.

So my question: is there any dimension which extends this notion further (negative numbers or even irrational numbers)? If so, what are the examples, how it can be used?

Salvador Dali
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See this discussion: https://physics.stackexchange.com/questions/52176/could-negative-dimension-ever-make-sense

Also, reals can include irrationals! So Hausdorff-Besicovitch already allows irrationals. And Mandelbrot himself has an interesting take on negative fractal dimension, where he says it measures the '"emptiness" of certain empty sets': http://users.math.yale.edu/~bbm3/web_pdfs/123negativeFractalDimensions.pdf

Community
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Brian Rushton
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The infinite lattice is a fractal of negative dimension: if you scale the infinite lattice on a line 2x, it becomes 2x less dense, thus 2 scaled lattices compose one non-scaled. If you take a lattice or on a plane, scaling 2x makes it 4x less dense so that 4 scaled lattices compose one non-scaled, etc.

Anixx
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