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.