Jellyfish, 20 bytes

p
<
)\&&&~j<i
->N>u0

Try it online! TIO can't handle the longer test cases, but given enough time and memory, they should work too. Explanation to come later.

Jellyfish, 20 bytes

p
<
)\&&&~j<i
->N>u0

Try it online! TIO can't handle the longer test cases, but given enough time and memory, they should work too.

Explanation

• i is input, and < decrements it. This value is fed to the function on the left.
• \> increments the value (at least once) until the function to the right gives a truthy value.
• The test function is a composition (by &s) of four functions.
• 0~j converts to string.
• u removes duplicate digits.
• > removes the head of the resulting string.
• N is logical negation: it gives 1 for an empty string, and 0 for non-empty. Thus the function tests for a rep-digit, and the result of \ is the next rep-digit counting from <i.
• )- subtracts the result from the function input, that is, <i.
• This difference is off by one, so < decrements it. Finally, p prints the result.