It is a common misconception that renormalization is needed only when infinities are coming up. But R. Shankar beautifully tackles this in his textbook on Quantum Field Theory and Condensed Matter in chapter 11th.
The central idea of renormalization is to do the computation by integrating out unnecessary mathematics, so we get good physics out of it. But a lot of good renormalization problems do not have anything to do with infinties. I will reproduce here one such example from the same book. Let us take a system $(a,b,c, \cdots,n; x,y)$, where $a,b,c,\cdots,n$ are parameters and $x,y$ are two variables. Calculating a partition function of such a system is easy. But what if we want to compute the partition function ignoring variable $y$. Such thing is achieved by writing a modified system $(a',b',c',\cdots,n')$
$Z(a',b',\cdots,n') = \int dx \left[ \int dy e^{-a(x^2+y^2)}e^{-b(x+4)^4} \right]$
$Z(a',b',\cdots,n') = \int dx e^{-S(a',b',c',\cdots,n')}$
where $S'$ is the action of the modified system. So
$e^{-S(a',b',c',\cdots,n');x} = \int dy e^{-a(x^2+y^2)}e^{-b(x+4)^4} \equiv \int dye^{-S(a,b,\cdot,n;x,y)}$
here we have created an effective action $S'$ of an effective theory. We did not eliminate $y$ by setting it zero, but we created an integral where we integrated out $y$ but got the same answer as the original theory. The second integral with integrated of exponential with Boltzman weight have interactions parts involving $x,y$. In the last we have modified the system in such a way that we do not need to figure out the coupling of $x$ and $y$, but we have set the fate of $x$ to itself.
