I'm finding various arithmetic functions satisfying this relation such that $f * f = f.$
Note that $*$ means a dirichlet multiplication (convolution). So, by definition of the dirichlet multiplication, we can see that $f * f = f$ would be $\sum_{d|n} f(d)f(n/d) = f(n)$ for some integer $n$. Specifically, we can see that $f(1) = 0$ or $1$ and $f(p) = 0$ or any number if $f(1) = 1$ for a prime $p$.
But, for the general number $n$, I could not find the arithmetic function satisfying this relation.