Finite Concepts that Require Infinity. Answers to this question, I hope, would address the objections one might have towards the (many different) foundations of infinity by examples of finite concepts - theorems, definitions, intuitions, perspectives, etc. - that require the infinite, whether that be in a proof, an important source of motivation, a key example (like some group like $S_3$ can be for non-abelian group theory), etc.
The examples-counterexamples tag could preempt criticism that the question is too broad.
What objections?
Well, for example, there're those that posit that infinity has no place in mathematics as, otherwise, we would be - and I quote - "quantifying the unquantifiable".
The Question:

What examples are there of finite concepts where infinity does some essential heavy-lifting?

Thoughts:
Nothing springs to mind.
An example might take the following form:

Theorem: (Only finite concepts hypothesised.) Lorem ipsum dolor sit amet, consectetur adipiscing elit, sed do eiusmod tempor incididunt ut labore et dolore magna aliqua. (Finite consequences.)

Proof: Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris nisi ut aliquip ex ea commodo consequat. (Infinity.) Duis aute irure dolor in reprehenderit in voluptate velit esse cillum dolore eu fugiat nulla pariatur. Excepteur sint occaecat cupidatat non proident, sunt in culpa qui officia deserunt mollit anim id est laborum. $\square$
(I used filler text but I hope this captures the sort of thing I'm looking for, without belabouring the point.)

What I want is a bunch of things to point to and say, "here: important finite things can require infinity" - and bring something to the table that is tangible in an area that is largely seen as intangible.
A Potential Example:
Finitely-presented groups can be infinite. For example, the presentation $$\langle a\mid \rangle$$ is finite, whereas the group $(\Bbb Z, +)$ it corresponds to is infinite.
Preliminary Discussion:
Before asking here, I brought it up in chat.
Please help :)
 A: Goodstein's Theorem qualifies as a theorem about the integers that can't be proved without using transfinite ordinals. Rather than attempt to summarize it here, I'll point to the Wikipedia essay linked in my comment on the original post, and also to some earlier discussions here on math.stackexchange:
Goodstein sequences
Do we know if there exist true mathematical statements that can not be proven?
Goodstein's sequences and theorem.
Counterintuitive Goodstein's Theorem
Goodstein's theorem
and a LOT of others.
A: In the sourcebook literature you can find Poincare's criticisms of Russell and Hilbert (Ewald's "From Kant to Hilbert" volume 2). Although his most important criticism is often taken to point to the distinction between predicative and impredicative mathematical reasoning, he also points out the circularity of trying to ground arithmetical induction.
This is now resolved with metatheory. It can be contrasted with the constructive mathematics of the Russian school. What the latter cannot represent will be the arithmetized mathematics that requires infinity in its assumptions. This probably cannot be made precise.
What clearly changes is the approach to quantification. In Hilbert's writing, "axiomatic" is  contrasted with "genetic." This reflects the backlash of nineteenth century mathematics against Kant's association of arithmetic with temporal intuition. But, it also relates to the Aristotelian distinction between "demonstrative" and "dialectical" reasoning. Aristotle views the succession of natural numbers in the manner Hilbert would call "genetic." Smaller numbers have "existential priority" with respect to larger numbers. And, earlier statements in a pedagogical demonstration are prior knowledge with respect to the statements which follow.
This is why the comparison with Russian constructive mathematics ought to be made.  Its feasible reasoning does not involve the usual notion of induction.
