Find the length of a number's "base-jumping" path
Jelly, 7 bytes

b@Ṁ‘ɗƬL


A dyadic Link accepting b on the left and N on the right which yields the sequence length.

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How?

b@Ṁ‘ɗƬL - Link: b; N
     Ƭ  - Collect up until no longer unique, applying:
    ɗ   -   last three links as a dyad - i.e. f(X[initially b], N):
 @      -     with swapped arguments:
b       -       convert (N) to base (X)
  Ṁ     -     maximum
   ‘    -     incremented
      L - length