Biological information encoded in genomes is fundamentally different from and effectively orthogonal to Shannon entropy. The biologically relevant concept of information has to do with ‘meaning’, i.e. encoding various biological functions with various degree of evolutionary conservation. Apart from direct experimentation, the meaning, or biological information content, can be extracted and quantified from alignments of homologous nucleotide or amino acid sequences but generally not from a single sequence, using appropriately modified information theoretical formulae. For short, information encoded in genomes is defined vertically but not horizontally. Informally but substantially, biological information density seems to be equivalent to ‘meaning’ of genomic sequences that spans the entire range from sharply defined, universal meaning to effective meaninglessness. Large fractions of genomes, up to 90% in some plants, belong within the domain of fuzzy meaning. The sequences with fuzzy meaning can be recruited for various functions, with the meaning subsequently fixed, and also could perform generic functional roles that do not require sequence conservation. Biological meaning is continuously transferred between the genomes of selfish elements and hosts in the process of their coevolution. Thus, in order to adequately describe genome function and evolution, the concepts of information theory have to be adapted to incorporate the notion of meaning that is central to biology.