Now also asked at MO:
Let $\sim$ denote the "no-disagreement" relation between partial functions: $f\sim g$ iff there is no $x$ such that $f(x)$ and $g(x)$ are each defined and are distinct.
Say that a differential field $\mathcal{K}=(K;0,1,+,\cdot,\partial)$ is fully concrete iff there is some way to assign, to each $\alpha\in K$, a partial function $\pi_\alpha$ on the constant subfield $C(\mathcal{K}):=\{\gamma\in K:\partial\gamma=0\}$ such that:
For $\gamma\in C(\mathcal{K})$, $\pi_\gamma$ is $\lambda x. \gamma$.
$\pi$ is injective-up-to-$\sim$, in the sense that $\pi_\alpha\not\sim\pi_\beta$ whenever $\alpha\not=\beta$.
$\mathcal{K}$ is closed-up-to-$\sim$ under composition, in the sense that for each $\alpha,\beta\in K$ there is a unique $\gamma\in K$ such that $\pi_\gamma\sim\pi_\alpha\circ\pi_\beta$. (Since the functions involved are partial, this falls short of $\pi_\gamma=\pi_\alpha\circ\pi_\beta$.) I'll write "$\alpha\circ\beta$" for this unique $\gamma$.
The field operations commute-up-to-$\sim$ with $\pi$, in the sense that $$\lambda x.\pi_{\alpha\star\beta}(x)\sim\lambda x.[\pi_\alpha(x)\star\pi_\beta(x)]$$ for $\star\in\{+,\cdot\}$.
The chain rule holds, in the sense that $\partial(\alpha\circ\beta)=[(\partial\alpha)\circ\beta]\cdot\partial\beta$.
Finally, say that a differential field is concrete iff it is a sub-differential field of a fully concrete differential field with the same constant subfield.
The difference between concreteness and fully concreteness takes Qiaochu Yuan's comment below into account: it's pretty clear that the differential field $\mathbb{C}(f)$ with $\partial(f)=f$ should be "concretizable" with $\pi_f=\lambda x.e^x$, but in order to get closure under composition we need to pass to a larger family of functions. We could have weakened the compositionality requirement instead - e.g. require only that whenever $\alpha,\beta,\gamma\in K$ have $\pi_\gamma\sim\pi_\alpha\circ\pi_\beta$ then there is a $\theta\in K$ with $\pi_\theta\sim\pi_{\partial\alpha}\circ\pi_\beta$ and $\partial(\gamma)=\partial(\theta)\cdot\partial\beta$ - but thinking in terms of "no-additional-constants" extensions seems more natural.
Concreteness seems like a natural property to consider for differential fields, but I haven't run into it before (I'm very new to differential fields though) and it's not obvious to me how common or rare it is. To get things started:
Suppose $\mathcal{K}$ is a differential field with the same cardinality as its constant subfield. Must $\mathcal{K}$ be concrete?
I'm also generally interested in any sources about concreteness, or any variant thereof - generally, when can a differential field be "faithfully" thought of as consisting of partial functions over its constant subfield?