Some things that would be useful to put in said script:
If the compiler has any types larger than size_t (useful for checking overflow) and/or [u]int_64_t. So check the size of [u]intmax_t, check if the compiler supports [u]int_128_t (and if so, what the type name is) - e.g. GCC does.
Related: if the compiler has any builtins for checking overflow / etc. GCC does, for instance.
If the compiler has a built-in static assert.
The various path operations. Notably, if we want to be able to cross-compile we can't just check "is this the case now".
Some things that would be useful to put in said script: