Is there any possibility that there exists a formula for the partial sums of the harmonic series? If not, why?

Question

I have been researching the harmonic series, $H_k$ and the Euler-Mascheroni constant, $\gamma$, quite a bit recently. I understand that the diagamma function has this property: $$\psi^{0}(k+1) + \gamma = H_k$$ And equivalently, $$\frac{d}{dk}[ln(k!)] + \gamma = H_k$$ My question is, does there exist some $f(k)$ such that $$f(k) + \gamma = H_k$$ where $f(k)$ does not require the use of special functions like $\psi$? This question is motivated by the fact that other sums have a formula for the partial sums, for example $$1+2+3+...+n=\frac{n(n+1)}{2}$$ I see how the sum of natural numbers up to $n$ is different than $H_n$; the former always has solutions in $\mathbb{N}$ and the latter (excluding $H_1$) always has solutions in $\mathbb{Q}$. Can $H_n$ have a coinciding formula only using elementary functions, and if not, why?