What are the most prominent uses of transfinite induction outside of set theory? What are the most prominent uses of transfinite induction in fields of mathematics other than set theory?
(Was it used in Cantor's investigations of trigonometric series?)
 A: Here's an example due to Erdos and Hajnal:
Theorem: There is a partition of plane into countably many pieces such that the distance between any two points in the same piece is irrational.
Corollary: Every non (Lebesgue) null subset $X$ of plane contains a non null subset $Y$ such that the distance between any two points of $Y$ is irrational.
Open question: Can we strengthen the corollary to: Every non null subset $X$ of plane contains a subset $Y$ of same outer measure as $X$ such that the distance between any two points of $Y$ is irrational?
Proof of theorem: By induction on $\kappa$, we show that 
$(\star)$: Whenever $X \subseteq \mathbb{R}^2$ has size $\kappa$, there is a well orering $\preceq$ of $X$ such that for every $x \in X$, the set of $\preceq$-predecessors of $x$ which are at a rational distance from $X$ is finite.
Note that $(\star)$ suffices to construct a rational distance free partition of $X$ into countably many subsets. 
When $\kappa \leq \omega$, this is obvious. So assume this is true for all $X \subseteq \mathbb{R}^2$ such that $|X| < \kappa$ where $\kappa \geq \aleph_1$. Let $|X| = \kappa$. Inductively construct $\langle X_i : i < \kappa \rangle$ such that the following hold:
(0) $X_i$'s are increasing continuous and $X = \bigcup \{X_i : i < \kappa\}$
(1) $|X_i| = \max(\aleph_0, |i|)$
(2) Whenever $x \neq y$ are from $X_i$, $z \in X$ and the distance of $z$ from each one of $x, y$ is rational, $z \in X_i$
Let $\preceq_i$ be a well order on $X_i$ witnessing $(\star)$. Define a well order $\preceq$ on $X$ as follows: If $x, y \in X_i \setminus \bigcup \{X_j : j < i\}$, then $x \preceq y$ iff $x \preceq_i y$. If $x \in X_i \setminus \bigcup \{X_j : j < i\}, y \in \bigcup \{X_j : j < i\}$, then $y \preceq x$. It is easy to check that $\preceq$ witnesses $(\star)$.
Komjath extended this to $\mathbb{R}^n$ for every $n$. The proof is slightly more complicated - $(\star)$ is replaced by a different statement which is again proved by transfinite induction (Note that $(\star)$ is false when $n \geq 3$).
A: Here are two contributions, one regarding complex analysis, the other number theory. But first some information to OPs question regarding Cantor and transfinite induction.
Historical Notes:
The following  is mainly based upon The Search for Mathematical Roots 1870-1940 by I. Grattan-Guinness. The author mentions in chapter 3: Cantor: Mathematics as Mengenlehre, that Cantor's remainder came into his hands in the late 1960s. Cantor's family asked him to set the papers in some order and then they were placed to the University of Göttingen.


*

*Über die Ausdehnung eines Satzes der Theorie der trigonometrischen Reihen (1872): This paper from 1872 shows his results regarding trigonometric series. He introduced the fundamental concept of limit-point (Grenzpunkt) of a set $P$. Then Cantor defined derived point-sets (abgeleitete Punktmengen) of $P$, each one comprising the set of limit-points of its predecessor. But he did not develop a line of thought regarding infinitieth derived sets $P^{(\infty)}$ at this stage.


According to I. Grattan-Guinnes Cantor's theory was doubtless a bit too intuitive at that stage. So, the concepts of well-ordered sets or transfinite induction were out of reach at that time.


*

*Über unendliche, lineare Punktmannichfaltigkeiten (1879):
Beginning with an extended study of the derivation of point-sets.

*Beiträge zur Begründung der transfiniten Mengenlehre (1897): This is the paper where ordinal numbers, limit-ordinals and well-ordered sets were defined and the principle of transfinite induction was used the first time.

Theorem by Paul Erdős
The following theorem by Paul Erdős was proved using transfinite induction.
Suppose, we have a family $\{f_\alpha\}$ of holomorphic functions  such that for each $z$ the set of values $f_\alpha(z)$ is countable. Let us denote this property $P_0$. Does it then follow that the family itself is countable?
The asthonishing answer is: it depends, namely on the continuum hypothesis.
Theorem (Erdős): If $c>{ \aleph_1}$, then every family $\{f_\alpha\}$ with property $P_0$ is denumerable. if $c=\aleph_1$, some family $\{f_\alpha\}$ with property $P_0$ has the cardinality $c$.
The original, short proof by Erdős can be found in a paper from 1963. The transfinite induction is an essential part of the proof in the case $c=\aleph{_1}$.

$$$$

Goodstein's Theorem
The following theorem about Goodstein sequences seems to be at the first glance purely number-theoretic, but it has some more interesting consequences.
Let $m,n$ be natural numbers $n> 1$. Write numbers $m$ as base $n$ representation of $m$ as follows. First write $m$ as sum of powers of $n$. For example, if $m=266$ and $n=2$ we obtain
  $$266=2^8+2^3+2^1$$
  Then write each exponent as the sum of powers of $n$ and repeat this process until the representation stabilizes.
  $$266=2^{2^{2+1}}+2^{2+1}+2^1$$
  Now define the number $G_n(m)$ as follows. If $m=0$ set $G_n(m)=0$. Otherwise set $G_n(m)$ to be the number produced by replacing every $n$ in the base $n$ representation of $m$ by $n+1$ and then subtracting $1$.
The Goodstein sequence for $m$ is defined by 
  $$m_0=m, m_1=G_2(m_0), m_2=G_3(m_1), m_3=G_4(m_2),\ldots$$
So, for example,
  \begin{align*}
266_0&=266=2^{2^{2+1}}+2^{2+1}+2\\
266_1&=3^{3^{3+1}}+3^{3+1}+2 \sim 10^{38}\\
266_2&=4^{4^{4+1}}+4^{4+1}+1\sim 10^{616}\\
266_3&=5^{5^{5+1}}+5^{5+1}\sim10^{10000}
\end{align*}
Now, despite the fact that some Goodstein sequences increase enormously fast at the beginning, the Goodstein's Theorem  surprisingly states that a Goodstein sequence for $m$ starting at $n$ eventually hits zero.
The proof is done using transfinite induction and the theorem is historically significant, since it was one of the first  presenting a true statement which is unprovable in Peano Arithmetic. This was shown in Accessible independence results for Peano arithmetic by Laurie Kirby and Jeff Paris in 1982. (Definition and example above are from this paper).
Here is a related MO answer.

A: Showing that there are exactly continuum many Borel subsets of the real line uses transfinite induction on some measure of complexity of the set relative to the open sets.
A: I think that Zorn's Lemma made it possible for mathematicians who are not familiar with set theory to use transfinite induction. For example, every commutative ring has a maximal ideal, you can prove that with transfinite induction, or use Zorn's Lemma. 
Learning transfinite induction, and the theory of ordinals, is a bit time consuming. Many mathematicians are perfectly fine by learning this simple one lemma, and avoiding to learn all else. 
Someone correct me if I am wrong, but before Zorn's Lemma mathematicians based their arguments on the transfinite. Since then hardly anybody does this anymore. 
A: By understanding transfinite induction by one of your answers earlier today, to this Question that I asked, I am also trying to understand the following result.
Let $\mathcal{E}$ be the class of elementary amenable groups, then using transfinite induction, we can give another description for the class of groups $\mathcal{E}$. Let $\mathcal{A}$ be the class of finitely generated abelian-by-finite groups and for each ordinal $\alpha$, define $\mathcal{L_\alpha}$ inductively as follows-
\begin{align}
\mathcal{L_0} & =\{1\} \\
\mathcal{L_\alpha}& =(L\mathcal{L_{\alpha-1}})\mathcal{A} & &  \text{if } \alpha \text{ is a successor ordinal},\\
\mathcal{L_\alpha} & =\bigcup_{\beta < \alpha}\mathcal{L_\beta} & & \text{if } \alpha \text{ is a limit ordinal}
\end{align}
Setting $\mathcal{L}=\bigcup_{\alpha\ge 0}\mathcal{L}_\alpha$, we can now state the following results.

$(\text{i} )\ \mathcal{L}=\mathcal{E}\\  \text{(ii)}\ \text{Each}\ \mathcal{L}_\alpha\ \text{is subgroup closed}$

This allows us to associate each $G\in \mathcal{E}$ with the least ordinal $\alpha$ such that $G\in \mathcal{L}_\alpha$.
In his paper Applications of a New $K$-Theoretic Theorem to Soluble Group Rings , Kropholler uses it as a lemma, along with many other results to settle Kaplansky's Zero Divisor Conjecture for soluble groups.
A: A promiment theorem that comes to mind is

Every vector space has a basis. More precisely, for every vector space $V$ over scalar field $F$ there is a (possibly very large!) basiss $B \subset F$ such that every vector $\mathbf{v} \in V$ can be represented uniquely as a finite sum $\mathbf{v} = \sum_{i=1}^k x_k \mathbf{b}_k$ with $x_1,\ldots,x_k \in F \setminus \{0\}$ and $\mathbf{b}_1,\ldots,\mathbf{b}_k \in B$. (Note $k$ will of course depend on $\mathbf{v}$).

This is shown by (in a sense) constructing $B$ explicitly. You simply add elements to $B$ until every $\mathbf{v}\in V$ can be represented, while taking care not to add any elements which would allow a vector to be represented by two different sums.
As long as the cardinality of $V$ is finite or countable, this is straight-forward induction, but to extend the theorem to arbitrary vector spaces, you need transfinite induction to formalize the proof. The easiest way of doing so is probably by using Zorn's lemma.
A: The small object argument is a very important technique based on transfinite induction allowing for the construction of approximations/factorizations in various algebraic and topological contexts.
A: The characterization of all real functions such that $f(x+y)=f(x)+f(y)$ for all $x,y\in\Bbb R$ relies on transfinite induction on a well-ordering of the real numbers. That question with its variations pops up here at this site from time to time.
A: A historically significant example is that of the Cantor-Bendixson rank which is in fact the result that motivated Cantor to invent transfinite induction, coming from the problem of describing the set where a Fourier series can converge/diverge.
A: There are several proofs of basic theorems of general topology which are proved by transfinite induction —for example, metrization theorems. You can find the very beautiful arguments explained in Hu's book on general topology.
