most irrational number

Today I was first exposed to the idea that, in a sense, we can say a number is "more irrational" than another... and even that there is a "most irrational number". The idea is to ask, "how well can we approximate the irrational with a simple rational?" We can approximate as close as we want, if we're willing to have huge numerators and denominators. But I can see that some irrationals will be approximated very well with a "small" fraction, and others won't. The ones that aren't are "more irrational".

This gets quantified by analyzing continuous fractions, and it turns out that ϕ (the golden ratio, or (1 + √5)/2) is the "most irrational". For more, see David Rutter's answer here.