Answer by Tom McFarlane:
First of all, it should be emphasized that a real number is either rational or irrational. Numbers are either one or the other, period, and both e and π are irrational.However, one can ask whether some irrational numbers are in some sense harder to approximate by rational numbers than others, and in that sense "more irrational." One way to make this notion precise is the, which assigns a positive number µ(x) to each real number x. Almost all transcendentals, and all (irrational) algebraic numbers have µ(x)=2, including e. But some transcendentals can have µ(x)>2. Liouville numbers, for example, have infinite µ(x). They can be approximated very well by rationals, while algebraic irrationals can not. So, counter-intuitively, the irrationality measure of an irrational is larger when it is better approximated by a rational. The irrationality measure of e is 2. The irrationality measure of π is not known. It is most likely 2, but could be larger.