How to tell whether you should bother reading this: Consider two well known facts about measurable functions: (i) the supremum of a sequence of measurable functions is measurable, (ii) if $f$ is measurable and $F$ is Borel then $F\circ f$ is measurable. It seems interesting to me that (i) is actually a special case of (ii). If you don't find that interesting you don't need to read this.
Teaching reals. I tell the kids that measurability is the last thing they should worry about, because although it may or may not be easy to prove, it never goes wrong in "real life". Then I give a more precise version, still too fuzzy to be susceptible to proof: If you start with countably many measurable functions and derive a function from them in any reasonable way the result will be measurable. (Commenting that of course the countability is important - measurability can go wrong in, say, probability, where we have an uncountable family of random variables to start with...)
I noticed something that has me wondering whether perhaps there is an actual theorem that includes the cases that actually come up.
Recall that if $X$ is a measurable space and $Y$ is a topological space then $f:X\to Y$ is measurable if the inverse image of every open set (equivalently every Borel set) is measurable. Hence for example if $X$ is a measurable space, $Y$ and $Z$ are topological spaces, $f:X\to Y$ is measurable and $F:Y\to Z$ is Borel then $F\circ f$ is measurable.
Question: Could it be that in the context of the last paragraph, if $F\circ f$ is measurable for every measurable $f:\Bbb R\to Y$ then $F$ must be Borel?
Regardless, it's clear that for example if $f,g:X\to\Bbb R$ are measurable and $F:\Bbb R^2\to\Bbb R$ is Borel then $F(f,g)$ is measurable.
What I just noticed that makes me think maybe there is a very general result about a function of countably many measurable functions being measurable is this: say $f_n:X\to\Bbb R$ for $n=1,2\dots$. Define $f:X\to\Bbb R^{\Bbb N}$ by $f(x)=(f_1(x),f_2(x),\dots)$. Give $\Bbb R^{\Bbb N}$ the product topology. Then $f_n$ is measurable for every $n$ if and only if $f$ is measurable.
In particular if $ (f_n)$ is a sequence of measurable functions and $F:\Bbb R^{\Bbb N}\to\Bbb R$ (or $[-\infty,\infty]$, whatever) is Borel then $F(f_1,f_2,\dots)$ is measurable. And this does include a lot of standard special cases.
For example the function $s:\Bbb R^{\Bbb N}\to(-\infty,\infty]$ defined by $s(x)=\sup_n x_n$ is Borel, being the limit of $s_n$, where $s_n(x)=\max(x_1,\dots,x_n)$. Hence (i) above is a special case of (ii).
Hence similarly for $\liminf$ and $\limsup$. And hence similarly for $f(x)=\lim f_n(x)$ if the limit exists, $0$ otherwise (because the characteristic function of the diagonal in $\Bbb R^2$ is Borel).
Raising the question of whether every example where $F(f_1,f_2,\dots)$ must be measurable arises from a Borel functionn $F$.
