Let $j$ be the Klein $j$-invariant (from the theory of modular functions).
Does $j$ satisfy a differential equation of the form $j^\prime (z) = f(j(z),z)$ for any rational function $f$?
No. Conceptually, the reason is that $j'(z)$ is a weakly holomorphic (= holomorphic except at the cusp at infinity, where it has a pole) modular form of weight $2$, so it cannot be expressed in terms of $j$ (weakly holomorphic modular form of weight $0$) and $z$ (not anywhere near being a modular form).
For a rigorous proof:
Note that $j(z+1) = j(z)$, so $j'(z+1) = j'(z)$.
Suppose that the $j$ invariant did satisfy a differential equation of your form. Then we'd have $f(j(z), z) = f(j(z+1), z+1) = f(j(z), z+1)$. Note that the functions $z$ and $j(z)$ are algebraically independent (this is just saying that $j(z)$ is a transcendental function). Hence the underlying two-variable rational function $f(x, y)$ satisfies $f(x, y) = f(x, y+1)$. This then easily implies that $f(x, y)$ must be independent of $y$, e.g. $f(x, y) = g(x)$ for some rational function $g$.
So our original differential equation must actually take the form $j'(z) = f(j(z))$. But the left hand side is a nonzero (weakly holomorphic) modular form of weight $2$ while the right hand side has weight 0, and a nonzero modular form has a unique weight, so this is impossible.
The j-function satisfies a third-order differential equation. I've learned this from an old paper of Daniel Bertrand which I am having trouble locating right now. Maybe is this one:
MR0550281 (81i:10042) Bertrand, Daniel Propriétés arithmétiques des dérivées de la fonction modulaire $j(\tau )$. (French) Séminaire de Théorie des Nombres 1977–1978, Exp. No. 22, 4 pp., CNRS, Talence, 1978, 10F37 (10F35)
(this is too long for a comment)
Here is the explicit equation of order three for the $q$-expansion of $j$ multiplied by $q$. Keep in mind that this does not prove that there is no order one differential equation, so it is not an answer to the question.
n [x ]f(x): 3 4 4 3 5 2 , 2 5 (2x f(x) - 6912x f(x) + 5971968x f(x) )f (x) - 2x f(x) + 3 4 4 3 6912x f(x) - 5971968x f(x) * ,,, f (x) + 3 4 4 3 5 2 ,, 2 (- 3x f(x) + 10368x f(x) - 8957952x f(x) )f (x) + 2 4 3 3 4 2 , 5 (6x f(x) - 20736x f(x) + 17915904x f(x) )f (x) - 6x f(x) + 2 4 3 3 20736x f(x) - 17915904x f(x) * ,, f (x) + 3 2 4 5 , 4 (x f(x) - 1968x f(x) + 2654208x )f (x) + 2 3 3 2 4 , 3 (- 4x f(x) + 7872x f(x) - 10616832x f(x))f (x) + 4 2 3 3 2 , 2 (5x f(x) - 8352x f(x) + 12939264x f(x) )f (x) + 5 4 2 3 , 5 4 (- 2f(x) + 960x f(x) - 4644864x f(x) )f (x) + 1488f(x) - 331776x f(x) = 0 , 2 3 4 f(x)= 1 + 744x + 196884x + 21493760x + O(x )]
A third-order differential equation for $j(\tau)$ is gotten via the Schwarzian derivative. The result is (1.13) of the paper by Harnad: http://arxiv.org/abs/solv-int/9902013