You get non-canonical representations such as 11=3

You could go multiplicative based. So 11 = 2×3. But then you get very difficult addition, comparison, and you need numerals inside your numerals. (E.g. 2048 would be 11 as in a single 11 in the first symbol spot)

It's true. I was teasing, but I did work this out once. IIRC it went something like:

  A = 1
  B = 2
  C = 3
  D = 5
  E = 7
  F = 11
  ... and so on

Then any number can be represented in terms of its prime factorization:

   4 : BB or B^B
   6 : BC
   8 : BBBB or B^C
   9 : CCC or C^B
  10 : BD
 ...
 100 : (BD)^B
 101 : # (some arbitrary unique symbol)
 ...

Actually base e is clearly the superior base as it's the most information-dense number representation.