How can we measure how "irrational" a number is?

Well, there is a sort of extension of the idea of irrationality called "transcendental". You can think of "irrational" as meaning "There is no way to relate this number to $1$ by thinking only about addition and subtraction". That is, a number $q$ is rational if we can write something like: $$q+q+q-1-1-1-1=0$$ or some other similar equation involving only $q,\,+,$ and $1$. The above represents the number $\frac{4}3$ and more generally, you can, for $q=\frac{a}b$ write: $$\underbrace{q+\ldots+q}_{b \text{ repetitions}}-\underbrace{1-\ldots-1}_{a \text{ repetitions}}=0.$$ or appropriately change signs for negative $a$.

The next logical step would be to consider the idea of allowing multiplication too - and this extends the rational numbers to algebraic numbers, which are (potentially) irrational, but can be related to the rationals via multiplication and addition. For instance, $x=\sqrt{2}$ satisfies $$x^2-2=0$$ and we can come up with similar expressions (the set of such equations would be polynomials to be precise) for any expression built with radicals and addition and such - and we could say that the degree of an algebraic number (i.e. how many multiplications we need) represents how irrational it is - that is $\sqrt{2}$ can be related to integers by squaring it, but $\sqrt[3]{2}$ requires cubing to get to an irrational - and numbers like $\sqrt{2}+\sqrt{3}$ require fourth powers. We might say that the degree represents a sort of "distance" from the rational numbers.

However, this only extends to a narrow class of numbers - $\pi$ and $e$ are both transcendental, meaning "not algebraic". We have much less understanding of these, since we can't relate them to the rationals through arithmetic - so we could be justified in saying that they are less well behaved then algebraic numbers. One way we can talk about their irrationality measure which essentially tells us how far from rational numbers a given irrational is, relating the growth of the denominator of the best rational approximations to how close they are - this represents how these numbers are comprehended by looking at sequences of rational approximations, rather than algebraic properties. However, this isn't terribly useful to compare actual numbers, because we hardly know how to calculate any. We do know that, from its infinite series, that $e$ acts very similarly to an algebraic number (looking at its series definition) and its irrationality measure is $2$. We don't know the irrationality measure of $\pi$, though we might suspect that it greater than $2$ - but it's hard to compare numbers this way, given our limited knowledge thereof, and all it means to have a small irrationality measure is that it's "far" from any rationals with small denominators.

A first thing to note is that there is a dichotomy among irrational numbers, some are roots of non-zero polynomials with rational coefficients, they are called algebraic numbers.

The others are called transcendental.

One has that $\sqrt{2}$ is algebraic, while $e$ and $\pi$ are not. In that sense $e$ and $\pi$ are perhaps more irrational.

There is a lesser known notion of periods, roughly things that can be expressed as integrals using only rational parameters. It is easy to see that $\pi$ is a period while it is unknown and doubtful that $e$ is a period.

In that sense $e$ is perhaps more irrational than $\pi$.

There is also such a thing as a measure of irrationality. It is known that the irrationality measure of every rational is $1$, of every non-rational algebraic number it is $2$, and it is at least two for transcendental numbers.

It is known that this measure is $2$ for $e$ while this is not known for $\pi$, though it might well be the case it is also $2$.

Long story short, yes, there are ways to further classify how irrational some number is, the details can however be complex and there are different ways to approach the subject.

There is another take on this that doesn't use the concepts of trancendentality or algebraity.

For any given irrational $a$, it's not hard to see that there exists a unique sequence of rationals $\{\frac{a_1}{1}, \frac{a_2}{2}, \frac{a_3}{3}, \ldots, \frac{a_n}{n}, \ldots\}$ such that each term is the closest rational to $a$ with that denominator.

Of course, we can then take a subsequence of terms $\sigma(k)$ such that $|a - \frac{a_{\sigma(k)}}{\sigma(k)}|$ is monotonically decreasing (arbitrarily, take the one that is biggest, in the sense of set inclusion).

We could say that irrational number $b$ is`more irrational' than $a$ if its corresponding monotonic sequence $\{|b - \frac{b_{\tau(k)}}{\tau(k)}|\}_{k \in \mathbb{N}}$ converges slower than $\{|a - \frac{a_{\sigma(k)}}{\sigma(k)}|\}_{k \in \mathbb{N}}$.

This is very related continued fractions - I haven't looked at the link APGreaves posted, but I presume it's similar.