Julia, 29 bytes

f(n)=n>9&&(f∘sum∘digits)(n)+1

Julia, 92 (29) bytes

f(n)=n>9&&(f∘sum∘digits)(n)+1

Edit: With correct printing it's 92:

f(n)=n>9&&(f∘sum∘digits)(n)+1
while(n=parse(Int128,readline()))≢π
println("$n ",f(n)%Int)end