Golden ratios or similar (there are a family of such ratios, it's not confined to just the Golden ratio) appear in nature often because of evolutionary selection pressure. Distribution of leaves around a stem, or flowers petals, or seeds are often optimised when the pattern used follows such irrational ratios.

e.g. if you want to maximise the light collected from one stem, then rotating the placement of leaves around the stem using such an irrational ratio will do this, or if you want to maximise the packing of seeds in a flower, then it represents a optimal design strategy.

If you think nature doesn't do irrational numbers, then you'll probably not want to know about how often *fractals* appear in nature as well. Just look at a fern leaf. Classic fractal pattern. Indeed, fractals are a brilliantly efficient way to encode a method of replication (which is why computer graphics use them all the time). The simplest of algorithms can lead to very complex and intricate patterns, as well as lead to optimal outcomes.

Then there are the appearance of prime numbers in nature as well, e.g. the number of years between when a species of cicadas appears in season is often a prime number of years (2,3,5,7,11,13,17). This evolved over millennia to be a very efficient survival strategy as it minimises the chances that two or more species emerge in the same season.

Another "natural" irrational number of course is ∏. ∏ appears everywhere in the natural world since spherical and related forms are a consequence of physics (e.g. gravity). The meandering ratio of rivers is a function of ∏. Biological processes are ultimately the consequence of physical laws, and is why ∏ appears in various places in nature. e.g. the patterns of spots and stripes on many animals has ∏ encoded within them. It's a key value in many periodic biological processes.