X+Y=Z - but in which base?
Jelly, 16 11 7 bytes

_/N,‘FṀ


This approach is heavily based on @beaker's Octave answer.

Input format is Z, Y, X, with little-endian digit order, using the digit 0 for unary.

Try it online! or run all test cases.

How it works

Rather than incrementally testing potential bases, this solves the polynomial that corresponds to the array P := X + Y - Z. This returns either the largest coefficient of P ≠ 0 – which has to be a root, since there is at least one valid base – or the highest digit of X, Y and Z, incremented by 1.

_/N,‘FṀ  Main link. Argument: [Z, Y, X]

_/       Reduce by subtraction; yield Z - X - Y.
         This works since Z must have at least as many digits as X and Y.
  N      Negate to yield X + Y - Z.
    ‘    Yield [Z, Y, X], with all digits increments by 1.
   ,     Pair the results to the left and to the right.
     F   Flatten the resulting, nested list.
      Ṁ  Compute the maximum.