Euler's method is generally applied as a numerical technique. But if you choose not to plug in values along the way, you can end up with a very long algebraic expression for $f(x)$ instead of a number. Then, you can take the limit of that expression as the step size shrinks to zero, and that should give you the general solution to the differential equation, in terms of the initial lower-order derivatives.
As an example, take:
$y'' = y'y\qquad y(0) = a\qquad y'(0) = b$
with step size $h$. So the method says to add $y'h$ to $y$, and $y'yh$ to $y'$, each step.
$\begin{array}{|c|c|c|} \hline x & y & y' \\ \hline 0 & a & b \\ \hline h & a + bh & b + abh \\ \hline 2h & a + 2bh + abh^2 & b + 2abh + a^2bh^2 + b^2h^2 + ab^2h^3 \\ \hline 3h & a + 3bh + 3abh^2 + a^2bh^3 + b^2h^3 + ab^2h^4 & ... \\ \hline \end{array}$
So the question is, is it possible to come up with a closed form for the continuum limit of the expression for $y$? Of course, since $h = \Delta x / n$, this will just be a power series in $\Delta x$, which we could alternatively get by just taking the derivative of the differential equation a bunch of times.
But I'm wondering if, by carefully following the combinatorics of how the terms for $y$ are generated via the operations of multiplication and addition, we could come up with an exact formula, however complicated, for that power series.
And if so, does that approach extend in principle to arbitrary differential equations (at least homogeneous ones), including coupled systems and even PDEs? Is there a combinatoric way of enumerating all the terms that would result from having arbitrary functions in the equation? Under what conditions would it be useful? Or is it inherently more complicated and unnecessary than other existing techniques?