Exponentiation (or tetration) is as much a symbol, it's just not explicitly penned down for brevity. The question could ask for a limitation on operators, but then I think it's outside the scope of the class.

However this is in essence a typographical question. A digit is a typographical entity! There is also a lot of implied machinery in writing 999 but for brevity's sake we omit it (base, writing system etc). If you allow symbols then this article comes into play http://www.scottaaronson.com/writings/bignumbers.html

If one permits exponentiation, then one would also permit tetration, because it's the typographic mirror. It also turns the question to one of radix informational efficiency in base 10 to one of typographical misinterpretation. Unfortunately, in this case, the typography is not interesting, because although I say implicit, I really mean "implicit". It's actually explicit. You can consider the the superscript here as functions taking inputs -- not digits, and thus is not a question of radix information efficiency.

Machinery is different from operators. We don't need to talk about other radices because digits explicitly means base-10.

Algebra is the math of symbolic logic, of objects and morphisms. I am not opposed to symbols, but rather the inclusion of operators. I also mention tetration as a cheeky answer, because I think it's as pedantic an answer as exponentiation.

Is ^ a digit?

9^9^9 is just the ASCII representation. When writing it 'on paper' you'd use superscripts which means you don't need anything other than digits.

That's because the symbolic operator is implied for brevity. If we allowed implicit operators, would tetration be permitted as well? It sounds like a distraction from the class subject.

Personally I'd allow it, but I'd also mark all those answers, (and any other reasonably clever answer) as correct on the test

As a testing company, maybe I would advise the teacher to give a discretionary, external reward for creativity, and maybe provide an advanced lecture for those inquiring students, but for policy reasons, people just want to know if students understood the content to measure pedagogical policy. You can either ask students to write an essay (unscalable and contingent on other skills), or you could ask a question with true and false answers.

Otherwise we cannot measure the efficacy of pedagogical policy.

He asked them to do that, since that is the right thing to do where a textual question can have multiple answers depending on the context; however they refused to do so.

That makes sense! I completely overlooked that.