Digitsum-based sequences

Question

I find Base-sensitive properties of numbers really interesting at times, and this is an example. For example, let f(x) be a recursive sequence where f(0)=2, and $f(x+1)=\digitsum(f(x)^n)$. If n equals 2, This sequence goes 2,4,7,13,16,13... and loops between 13 and 16. Starting with a different value, like f(0)=17, the sequence goes 17,19,10,1,1,1... . okay, one more time, let f(0)=13, and let's let n equal 3 this time. The sequence now becomes 13,19,28,19... . My question is, is it possible for any starting number x and power n to cause an infinite sequence that tends to infinity and doesn't form a loop?