With casimir effect, you could modify the though experiment so as to avoid infinities (while using renormalization).
For instance, instead of considering $2$ (conducting) plates $D_1,D_2$ with distance $d$ apart, you can consider 4 plates $D_0,D_1,D_2,D_3$, with distance $(D_0,D_1) = (D_2,D_3) = L $. After having renormalised modes, for instance, in one dimension, you will have, for energy between two plates due to internal modes :
$$E(\epsilon, l) = \frac{1}{2} \sum_{n=1}^{+ \infty} \frac{n \pi}{l}~e ^{- \large \epsilon ~(\large \frac{n \pi}{l})}$$
The force between the 2 plates $D_1,D_2$, due to internal modes, is $f(\epsilon,l) = - \frac{\partial E(\epsilon,l)}{\partial l}$
To calculate the total force between the 2 plates $D_1,D_2$, you have to substract the force due to the internal modes $D_0,D_1$ or $D_2,D_3$, so the total energy is :
$$f_{CASIMIR}(d, L) = lim_{\epsilon \rightarrow 0} ~~(f(\epsilon, d) - f(\epsilon, L))$$
The term $f(\epsilon, d) - f(\epsilon, L)$ does not contain infinite terms in $\frac{1}{\epsilon}, \frac{1}{\epsilon^2}, etc...$, so there is no problem.
No, we are going to send the plates $D_0,D_3$ at infinity, this is the
limit $lim_{L \rightarrow + \infty}$ . This means simply that these plates , in some sense, don't exist any more.
So, finally :
$$f_{CASIMIR}(d) = lim_{L \rightarrow + \infty} ~~ lim_{\epsilon \rightarrow 0}~~ (f(\epsilon, d) - f(\epsilon, L))$$