Q:

Is there a problem for which it is known that the only solution is "iterative"?

By "iterative solution" I mean specifically the following type of iteration: given a problem whose solution is $x$, first you compute some approximate solution $x_n$, and then make use of $x_n$ to find $x_{n+1}$; and furthermore the sequence $\{x_n\}$ converges to $x$.
If $x_{n+1} = x_{n} + f(n)$ for a known $f$, then clearly $x_{n+1} = \sum_0^n{f(n)}$, and thus the computation of $x_n$ requires merely a knowledge of $f(n)$ and would not be "iterative" according to my definition.
Consider as an example the equation $x^2 - 2 = 0$.  Newton's method applied to this equation is "iterative", but there also exists an infinite series for $\sqrt{2}$. 
Question: Is there a problem for which it can be proven rigorously that the only possible solution is "iterative"?
(Apologies for not defining "iterative" more rigorously, any suggestions for a better definition would be appreciated)

A:

Ultimately, I don't think "iterative" is a well-defined concept. As you note, if we have $x_{n+1}=x_n+f(n)$ then the limit is $x_0+\sum_{i=0}^{\infty}f(i)$, which, even if computing $f(i)$ would be much easier given $x_i$ than not, is certainly a well-defined infinite series - and ultimately, distinguishing between an infinite sum that yields a given value and the limit of a sequence of partial sums is not meaningful, since the former is typically taken to be the latter.
To be sure, typically, one defines every irrational number as the limit of some sequence, and the irrational cannot be obtained any other way - this is because we can easily define the rationals, and then consider that irrational numbers are just limits of Cauchy sequences of rationals (i.e. the reals are the completion of the rationals), so you could say that every irrational number is the result of an iterative process.
It's also somewhat unclear what it means to be a "solution". Like, we can talk about the algebraic properties of $x=\sqrt{2}$ without knowing where on the number line it falls because it is an algebraic number. Like, we know that $x^2=2$ and $x^{-1}=\frac{x}2$ and things like that even without knowing that $\sqrt{2}$ is somewhere in $(1,2)$ - and entire branches of mathematics, like Galois theory work by treating $\sqrt{2}$ as if it were similar to extending the reals by $i$. Even without having any idea where $x=\cos(x)$ is solved, one can make statements like, for that $x$, it must be that $\sin(x)^2=1-x^2$, where we just consider the answer as a hypothetical number satisfying a given property and discuss what is true of it. 
Finally, it is often possible to answer the question "Is $x$ in the interval $[a,b]$?" without invoking limits - like, if $a^2<2$ and $b^2>2$, then $\sqrt{2}$ is definitely somewhere in between. This, unlike an infinite series of limits, where we do not intrinsically know how close a partial sum is to the answer, gives a tad more certainty about the value of $x$ than any technique involving infinitely many terms.

