How Persistent am I?

Python 2, 50 39 bytes

-10 thanks to @JonathanAllan

f=lambda x:x>9and-~f(sum(map(int,`x`)))

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Brachylog, 10 bytes

;.{ẹ+}ⁱ⁾Ḋ∧

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Explanation

;.{  }ⁱ⁾     Iterate Output times on the Input…
        Ḋ∧   …so that the result is a single digit:
   ẹ+          Sum the elements

Prolog (SWI), 91 90 bytes

s(A,B):-A=0,B=0;divmod(A,10,Q,R),s(Q,T),B is T+R.
p(A,B):-A<10,B=0;s(A,S),p(S,T),B is T+1.

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