How about this one: $f(n)=1$ if $2^{2^n}+1$ is prime, and $f(n)=0$ otherwise.
Or: $f(n)$ is the largest prime factor of $2^{2^n}+1$.
Added 1: Inspired by Barry Cipra's comment, here is a function which is unknown in principle, not just in practice: Let $r(m)$ denote the number of lattice points on the sphere $x^2+y^2+z^2=4m+1$, and define $f(n)$ as the largest $m$ with $ r(m) < n\cdot m^{1/3}$.
Added 2: Here is another function, inspired by Barry Cipra's example: $f(n)$ is the $n$-th positive integer which is a sum of three cubes (including negative cubes).
