69th Belarusian Mathematical Olympiad

Answer: yes. Note that if function $f(x)$ satisfy $f(f(n))=2^{n-1}$ for any positive integer $n$, then it satisfy the problem condition $f(f(n+1))=f(f(n))+2^{n-1}$ as well. It is well known that there exist infinitely many functions $f:\mathbb{N}\to \mathbb{N}$ such that $f(f(n))=2^{n-1}$ for all positive integers $n$.