What are some examples of fixed point problems where transfinite constructions elucidated the problem? Maths is filled with various different kind of "fixed point theorems", sometimes even when they are not phrased as such, where we have an operation $f$ on objects of a certain kind and wish to find an object $X$ such that $X = f(X)$. Methods of proof vary but many can be essentially proved by something along the lines:
"Start with any object $X$, apply $f$ repeatedly and as you go agglomerate the images of $X$ together in some way. Then if you do this enough times and maybe take a limit and you'll reach a fixed point."
Some examples of what I'm talking about are:

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*Let $\omega$ be some $n$th root of unity. Is there a number $z$ in $\mathbb{C}$ such that $\omega\cdot z = z$? Well we could just start with 1 and then $\omega\cdot 1 = \omega$. Then we "agglomerate" to get $1+\omega$ which might not work so we multiply by $\omega$ to $\omega + \omega^2$... Doing this enough times gives us the expression $1+\omega + \ldots + \omega^{n-1}$ which is invariant under multiplication by $\omega$, since $\omega \cdot (1+ \omega + \ldots + \omega^{n-1}) = \omega + \omega^2 + \ldots + \omega^{n-1} + 1$. Indeed, this number is zero (as could actually be seen by thinking about the equation directly but this is illustrative of the method).

*Suppose we want a function $f:\mathbb{R}\to\mathbb{R}$ such that its derivative is equal to itself, i.e. $f' = f$. Here we could start with any polynomial and then eventually get $0$, which works. Alternatively, looking for a more interesting situation, we could actually work backwards and ask for a function whose derivative is zero. Say we pick $1$, perhaps out of simplicity, and then again move backwards to get $x$ and then again to get $\frac{x^2}{2}$ etc. and agglomerating all these we get $1, 1+x, 1+ x + \frac{x^2}{2},\ldots$ which some sense of a limit becomes $e^x = 1+x + \frac{x^2}{2} + \ldots$

*Suppose we want to find an even function, i.e. $f:\mathbb{R}\to\mathbb{R}$ such that $f(x) = f(-x)$ for all $x \in\mathbb{R}$. Start with any function $f_0$ at all, perform the "evenness map" to it to form $f_1$ defined as $f_1(x) := f_0(-x)$ and then agglomerate it with the original function to form $f_1 + f_0$, which is even. Slight variants on this reasoning are how you prove any real function is the sum of an even and an odd function, how you prove that every square matrix is the sum of a symmetric and skew-symmetric matrix and how you prove that every function $f:[a,b]\to\mathbb{R}$ of bounded variation is the difference of two monotone functions.

I could go on to give more examples (every normal function on the ordinals has a fixed point, etc.) of this proof technique but hopefully it's clear. What I wish to ask about is when this kind of process was made to work by using transfinite iteration.
I know of essentially only one application of this: the automorphism tower problem for groups. The problem of trying to understand the long-term behaviour of the sequence $G, \text{Aut}(G), \text{Aut(Aut(}G)),\ldots$ for a given group $G$ was posed some time ago and people were interested in knowing if every such sequence eventually became constant regardless of the starting $G$. Various pieces of work were done until Joel David Hamkins came and proved that it was most natural to extend the automorphism sequences transfinitely, at which point it was clear that yes every such sequence terminates.

Sorry for the long build up but the question ultimately is:
Are there other examples of such transfinite iteration being used to find a fixed point or more generally construct some important object?
 A: There is the Sadovskij construction for a map $f\colon K\to X$ with $K$ being a subset of a topological vector space $X$.
$K_0:=K$, $K_{\omega+1}:=\overline{\mathop{conv}} f(K_{\omega}\cap K)$, and $K_\lambda:=\bigcap_{0<\omega<\lambda}K_\omega$ if $\lambda$ is a limit ordinal.
Then $K_\lambda$ is eventually constant $K_\infty$, and in particular $f(K_\infty\cap K)\subseteq K_\infty$; if $K_\infty$ is nonempty and compact and contained in $K$ and $X$ is a locally convex space and $f$ is continuous on $K_\infty$ then $f$ has a fixed point by Schauder's fixed point theorem. (The non-emptyness can be ensured by a slight modification of the construction, and the compactness holds if $f$ is a so-called “condensing” map, that is, if for every set $M$ the image $f(M)$ is “more compact” than $M$). Similar constructions work also for multivalued maps and in the more general context of degree theory (when $K_\infty$ is in general not a subset of $K$).
However, the transfinite induction can usually be avoided by considering the intersection $M_\infty$ of all closed convex sets $M$ satisfying $f(M\cap K)\subseteq M$ (in general, $M_\infty\subseteq K_\infty$). Moreover, under very mild assumptions, it can already be proved that $K_\lambda$ is already compact for the first infinite ordinal $\lambda=\omega_0$.
The original paper about the so-called ultimate range $K_\infty$ is
B.N. Sadovskii, Limit-Compact and Condensing Operators (in Russian), Uspekhi Mat. Nauk 27 1972 (1), 81-146.
The English translation appeared in Russian Math. Surveys 27 1972 (1), 85-155.
The sets $M_\infty$ are usually called fundamental sets and were introduced by M.A. Krasnoselskii, P.P. Zabrejko, or V.V. Obukhovskii - I really do not know which is the first reference here.
The fact that $K_{\omega_0}$ is at the first infinite ordinal cocompact for a much larger class of maps than one might expect at a first glance is highly nontrivial and was perhaps first observed by myself in M. Väth, Fixed point theorem and fixed point index for countably condensing maps, Topol. Methods Nonlinear Anal. 13 1999 (2), 341-363.
Meanwhile, fundamental sets are a standard approach to define a degree theory for non-compact maps. As mentioned, the approach works for large classes of multi-valued maps as well as for coincidence points, and actually even combinations of both.
