The geometric mean can be thought of as the exponential of the arithmetic mean of the logarithms of your dataset $\{a_i\}_{i=1}^n$: $$GM = \exp\left(\dfrac{1}{n}\sum_{i=1}^{n}\ln(a_i)\right)$$
Similarly, the standard deviation of a dataset is the square root of the arithmetic mean of the squares of your errors $\{e_i\}_{i=1}^{n}$: $$SD = \sqrt{\dfrac{1}{n}\sum_{i=1}^{n}(e_i)^2}$$
I find it interesting that both of these ideas seems to follow of more general pattern: $$f^{-1}\left( \dfrac{1}{n} \sum_{i=1}^{n} f(x_i) \right),$$ where the function in question is $f(x) = x^2$ for the standard deviation, and $f(x) = \ln x$ for the geometric mean. Even the arithmetic mean is trivially of this form, just with $f(x)= x$ as the identity function.
Are there other widely-used variations of these kinds of "functional" averages? And is there anything we can say, more universally, about these kinds of averages as a whole?
In particular, I find it interesting that all three of the averages I mentioned above (when applied to two values) all give values that are between the two data points. What would have to be true about a function $f(x)$ for this "midpoint" property to hold? For example, $\arcsin\left( \frac{1}{2} (\sin a + \sin b)\right)$ is most certainly not between $a$ and $b$ in most cases.
This type of average is called a quasi-arithmetic mean when $f$ is continuous. The midpoint property is then guaranteed because $f$ must be strictly monotonic if it is continuous and has a left inverse $f^{-1}$.
As you observed, choosing $f(x)=\ln(x)$ (or $f(x)=\log_a(x)$ for any positive $a \neq 1$) results in the geometric mean. Another notable example is that $f(x)=\frac{1}{x}$ corresponds to the harmonic mean.
For any such $f$, the $f$-mean $M_f(\vec{x})=f^{-1}\left(\frac{1}{n} \sum_{i=1}^n f(x_i)\right)$ enjoys many other properties that would be expected of an averaging function (see the Wikipedia page). It is easy to show, for instance, that the $f$-mean has idempotency in that for all $x$, $M_f(x,\cdots,x)=x$.