Find the longest rep digit
Jelly, 9 bytes

b‘Ḋ$EÐfZL


A monadic link accepting and returning numbers

Try it online! or see a test suite (inputs one to 32 inclusive).

How?

b‘Ḋ$EÐfZL - Link: number, n
   $      - last two links as a monad:
 ‘        -   increment = n+1
  Ḋ       -   dequeue (with implicit range build) = [2,3,4,...,n+1]
b         - convert to those bases
     Ðf   - filter keep if:
    E     -   all elements are equal
       Z  - transpose
        L - length (note:  length of the transpose of a list of lists is the length of the
          -                longest item in the original list, but shorter than L€Ṁ)




...or maybe I should have done:

bḊEÐfZLo1


For the Lo1z.