Is $\pi$ more transcendent than $e$?

Solution 1:

This question at MathOverflow has several answers discussing several senses in which one can say that one real number is more irrational than another, including irrationality measures and other hierarchies of complexity for real numbers.

Solution 2:

There are different kinds of irrationality measures:

http://mathworld.wolfram.com/IrrationalityMeasure.html

As you can see in the table above $e$ is not more irrational than algebraic numbers, but it is not clear if $\pi$ is more irrational, but both are not very irrational compared to Liouville numbers.

Solution 3:

You might be interested in Mahler's classification of transcendental numbers, which you can start to read about at http://en.wikipedia.org/wiki/Transcendental_number#Mahler.27s_classification