Allocator named requirements -- exceptions

Any limitations on "appropriate" implied elsewhere?

No. "Appropriate" qualifier has no objective meaning. It's effectively a suggestion to use common sense. There are no limitations on the type thrown from allocate.

Can a valid custom allocator just throw a double on any request?

It would be conforming if the author considers double to be appropriate. I wouldn't consider it "appropriate" myself, but it's up to the author to decide.

Context: implementation of a noexcept function that uses allocator, but has fallback strategy to do something if all allocations fail.

You should use a catch-all block:

try {
    ptr = a.allocate();
} catch(...) {
    // deal with it
}