Do harmonic numbers have a “closed-form” expression?

There is a theory of elementary summation; the phrase generally used is "summation in finite terms." An important reference is Michael Karr, Summation in finite terms, Journal of the Association for Computing Machinery 28 (1981) 305-350, DOI: 10.1145/322248.322255. Quoting,

This paper describes techniques which greatly broaden the scope of what is meant by 'finite terms'...these methods will show that the following sums have no formula as a rational function of $n$: $$\sum_{i=1}^n{1\over i},\quad \sum_{i=1}^n{1\over i^2},\quad \sum_{i=1}^n{2^i\over i},\quad \sum_{i=1}^ni!$$

Undoubtedly the particular problem of $H_n$ goes back well before 1981. The references in Karr's paper may be of some help here.

Harmonic numbers can be represented in terms of integrals of elementary functions: $$H_n=\frac{\int_0^{\infty} x^n e^{-x} \log x \; dx}{\int_0^{\infty} x^n e^{-x} dx}-\int_0^{\infty} e^{-x} \log x \; dx.$$ This formula could also be used to generalize harmonic numbers to fractional or even complex arguments. These generalized harmonic numbers retain some of their useful properties, for example, $$H_z=H_{z-1}+\frac{1}{z}.$$

This is probably not what you were looking for (since it isn't in terms of rational or elementary functions), but for the harmonic numbers we have

$$H_n=\frac{1}{n!}\left[{n+1 \atop 2}\right]$$

where $\left[{n \atop k}\right]$ are the (unsigned) Stirling numbers of the first kind (page 261 from the book Concrete Mathematics by Graham, Knuth and Patashnik - second edition).

For the generalized harmonic numbers I like this formula - even though it does involve an integral and Riemann zeta...

Maybe you prefer this