Calc of max, or max of calc in CSS

A 'pure' css solution actually is possible now using media queries:

.yourselector {
  max-width: calc(100% - 80px);
}

@media screen and (max-width: 500px) {
  .yourselector {
    max-width: 500px;
  }
}

min(), max(), and clamp() are finally available!

Support starts in Firefox 75, Chrome 79, and Safari 11.1 (clamp is available in Safari 13.1).

min() and max() take any number of arguments.

clamp() has syntax clamp(MIN, VAL, MAX) and is equivalent to max(MIN, min(VAL, MAX)).

min() and max() may be nested. They can be used in calc() as well as outside of it, they also may contain math expressions, that means you can avoid calc() when using them.

Therefore the original example can be written as:

max-width: max(500px, 100% - 80px);

No you cannot. max() and min() have been dropped from CSS3 Values and Units. They may be re-introduced in CSS4 Values and Units however. There is currently no spec for them, and the calc() spec does not mention that either are valid inside a calc() function.