Or, how does the value of max input signal that I can feed to the op amp changes with the supply voltage?
I hope you understand this is opamp-specific question.
First distinction we must make is the chapter named:
a) Absolute maximum rating and
b) Operating conditions
Chapter (a) describes how to not disable the device permanently. Once you violate those rules - there is no need to read chapter (b).
Conditions from chapter (b) describe where the device behavior is qualified.
So if chapter (a) starts from: "Thou shalt not put more than Vdd on input pins." then just don't. But that sentence does not say that if you apply Vdd-1V then the opamp would amplify at all.
Such details are described in chapter (b) where it is written that: "At 25degC, Vdd-Vss=5V, common mode input voltage range that we tested is 0V up to Vdd-1.5V" for example. Then you can expect input stage to operate as intended.
Of course that sentence does not say that the inverting amplifier arrangement with +1V input would output -1V. The output stage operation is characterized with the output swing parameter (and also some plots usually). Most likely it won't output anything below Vss.