APL (Dyalog Unicode), 28 bytes

{⌈/≢¨⍵/⍨(1=≢∘∪)¨⍵}⊢⊥⍣¯1⍨¨1+⍳ 

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Pretty straightforward implementation.

{⌈/≢¨⍵/⍨(1=≢∘∪)¨⍵}⊢⊥⍣¯1⍨¨1+⍳ 1+⍳ ⍝ Make a range [2, n + 1] ¨ ⍝ For each b in that range ⊢⊥⍣¯1⍨ ⍝ Interpret n in base b (gives a vector of integers) ⍵/⍨ ⍝ Keep the representations where (1=≢∘∪)¨⍵ ⍝ it's a repdigit ¨⍵ ⍝ For every representation, ≢∘∪ ⍝ is the length of it, without duplicates 1= ⍝ equal to 1? ≢¨ ⍝ Get the length of each of the remaining representations ⌈/ ⍝ Get the biggest length 

APL (Dyalog Unicode), 28 26 bytes

Saved 2 bytes thanks to Adám!

{⌈/≢¨⍵/⍨1=≢∘∪¨⍵}⊢⊥⍣¯1⍨¨1+⍳ 

Try it online!

Pretty straightforward implementation.

{⌈/≢¨⍵/⍨(1=≢∘∪)¨⍵}⊢⊥⍣¯1⍨¨1+⍳ 1+⍳ ⍝ Make a range [2, n + 1] ¨ ⍝ For each b in that range ⊢⊥⍣¯1⍨ ⍝ Interpret n in base b (gives a vector of integers) ⍵/⍨ ⍝ Keep the representations where 1=≢∘∪¨⍵  ⍝ it's a repdigit ¨⍵ ⍝ For every representation, ≢∘∪ ⍝ is the length of it, without duplicates 1= ⍝ equal to 1? ≢¨ ⍝ Get the length of each of the remaining representations ⌈/ ⍝ Get the biggest length