Is there an example of a Radon-Nikodym derivative on $(\mathbb R, \mathcal B)$ that is not a classical derivative? Is there an example for some two measures $\mu, \nu$, of a Radon-Nikodym derivative on $(\mathbb R, \mathcal B)$ that is not a classical derivative?
Edit: @drhab gave a good answer, and here is a follow up question. If we define for $\mu$ and $\nu$ their cumulative probability functions (thereby assuming they are probability measures). That is, what if $\mu([a,b])=F(b)-F(a)$ and $\nu([a,b])=G(b)-G(a)$.
Are there then cases where the Radon-Nikodym derivative exists, but the standard $\epsilon, \delta$ derivative does not? 
 A: If $\mu$ and $\nu$ are Borel probability measures on $\mathbb{R}$, then it is always true that there are right-continuous non-decreasing functions $G$ and $F$ such that $F(b^{+}) - F(a^{+}) = \mu((a,b])$ and $G(b^{+}) - G(a^{+}) = \nu((a,b])$.  Therefore, I'm not sure what the intention of your edit is.  This is not a special case and doesn't change the answer to your question.
At any rate, the only time we should expect $\frac{d \mu}{d \nu}(x) = \lim_{y \to x} \frac{F(y) - F(x)}{y - x}$ generically is when $\nu$ is Lebesgue measure.  In fact, for $\nu$-a.e. $x$, we have the formula
$$\frac{d \mu}{d \nu}(x) = \lim_{\epsilon \to 0^{+}} \frac{\mu((x - \epsilon,x + \epsilon))}{\nu((x - \epsilon,x + \epsilon))} = \lim_{\epsilon \to 0^{+}} \frac{F(x + \epsilon) - F(x - \epsilon)}{G(x + \epsilon) - G(x - \epsilon)}.$$ 
(This is a consequence of the Besicovich differentiation theorem.)
Therefore, the only case when $\frac{d \mu}{d \nu}$ is equivalent to $\frac{dF}{dx}$ independently of $F$ is when $\nu$ is Lebesgue mesaure.  In some sense, this explains why we sometimes write $dx$ for Lebesgue measure, and it also justifies the notation $\frac{d \mu}{d \nu}$.
A: Let $\mu$ be defined by $\mu(B)=|\mathbb Z\cap B$|. 
Let $\nu$ be defined by $\sum_{n\in\mathbb Z\cap B}f(n)$ where $f:\mathbb Z\to[0,\infty)$.
Then $\nu<<\mu$.
Prescribe $g:\mathbb R\to[0,\infty)$ by $x\mapsto f(x)$ if $x\in\mathbb Z$ and $x\mapsto0$ otherwise.
Then $g$ serves as Radon-Nikodym derivative of $\nu$ wrt $\mu$.
Does it meet your demand of being a "non-classical derivative"?
