Can we, in a certain way, quantify the measure of non-differentiability of functions that are continuous everywhere but differentiable nowhere? I am not sure how to ask this question because it seems to me that my thoughts on this topic are not clear enough, but I will give it a try.
What, really, do I want to know?
Well, I would like to know is there any "measure" on how "far" is such a function from being differentiable at some point?
In other words, and hopefully, more precise ones:

Suppose that $f$ is some function that is continuous at every point of its domain and differentiable nowhere. Is there a way to assign to every point of such a function a number which measures "how far is function from being differentiable" at that point, or, "how non-differentiable" the function at that point is? 

I like to believe that some of you really understand the spirit of the question and what it is all about, although I at the moment do not know how to state it more correctly and more rigorously.
 A: Well, remember that the derivative of $f$ at $c$ is just the limit of the difference quotients $$D_c^f(h)={f(c+h)-f(c)\over h}$$ as $h$ goes to zero (this notation is not standard). If the limit $\lim_{h\rightarrow 0}D_c^f(h)$ does not exist, this is because $D_c^f$ behaves "wildly" around $h=0$. So, one thing we can do is ask: how wild is the behavior of the difference quotient near $h=0$? There are a few ways we might formalize this, but the most immediate (to me at least) is:

Say that the wildness of $f$ at $c$ is $$wild_f(c)=\lim_{k\rightarrow 0}(\sup\{\vert D_c^f(h_0)-D_c^f(h_1)\vert: 0<\vert h_0\vert, \vert h_1\vert<k\}).$$

This wildness is measuring how badly the difference quotients of $f$ fail to converge as we make $h$ approach $0$. Then the overall wildness of $f$ might be $\sup\{wild_f(c): c\in dom(f)\}$.
As a good exercise, for any $\epsilon>0$ you can find a function $f$ which is nowhere differentiable, but such that $wild_f(c)<\epsilon$ for every $c\in\mathbb{R}$. You can also find a function $f$ such that for every $c\in\mathbb{R}$, $wild_f(c)=\infty$. 
I'm not sure this is actually useful for anything, but is is a definition which seems to capture your intuitive question, and which makes sense and is nontrivial.
A: You might want to consider Hölder continuous functions: The higher the Hölder order $\alpha\in [0,1]$ of a continuous function, the more regular the function is. The problem with this idea is that for $\alpha=1$ you won't get a differentiable function, but only a Lipschitz continuous function, and to the best of my knowledge there is now way to parametrize the further step from Lipschitz to differntiable. However, it is true that by Rademacher's theorem Lipschitz continuous functions are differentiable almost everywhere, so Hölder's theory may help if you relax your requirement and content yourself with characterizing "global" differentiability instead of differentiability at a given point.
A: My feeble memory
seems to recall a 
discussion somewhere
that concluded that
almost all continuous functions 
behave "badly",
that is,
no derivatives anywhere.
I vaguely recall
"Baire category"
being there,
or some other
measure theoretic terms
I don't really understand.
This should probably be a comment,
but,
since it sort of answers the question,
but not really,
I am making it
an answer.
