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Taking the principal log of a real or complex number an infinite number of times converges one of two particular values in the complex plane. These values are $-W(-1)^*$ given a seed value with $Im(z) ge 0$, and $-W(-1)$ otherwise (with $W$ being the principal branch of the Lambert W function.)

While converging these values appear to "spiral" around this number in a way reminiscent of certain systems of differential equations.

However, function application is a discrete operation, so relating the two might be a bit of a fool's errand. I'm aware that it is possible to extend the differentiation operator in this manner. But that may be specific to differentiation, and it might not be possible with general function application.

So, is there a method for a general function? What about the specific function log? Is there a class of functions for which this works?

The simplest case of fractional iteration is when $,f(x) := ccdot x,$ with fixed point $0$, convergence is linear and the converging factor $,0<c<1.,$ Then the fractional iteration $,f^{(t)}(x) = c^tcdot x.,$ Complications arise if $c$ does not satisfy the bounds. That is, the exponentiation $,c^t,$ is multivalend. The general case reduces to the simplest case using conjugation of composition with the convergence factor still the first derivative at the fixed point. Similarly for quadratic or higher orders of convergence.

In the example of $,f(x):=log(x),,$
$,f(w + x) approx w + c,x,$ where $,wapprox 0.318+1.337i,$ and the convergence factor $, c := f'(w),$ is
$approx 0.168-0.707i.,$ Now we want an expansion for $g(x)$ so that
$$log(w+g(x)) = w+g(ccdot x)$$ where the conjugating function
$g(x)$ has a power series expansion
$$g(x) approx x + (-0.151 -0.296i)x^2 +(-0.036+0.098i)x^3 + (-.025-0.017i)x^4 + O(x^5).$$

It's called fractional iteration. There are papers on it and some research, among other things in the domain of fractals, but it's not a widely known subject. There are quite some restrictions on what properties the functions needs to have. The simplest case is to consider behaviour of a function between two stationary points (intersect the $f(x)$ and $x$ curves). On every such a bump, between two stationary points (assuming the function there is also monotonic), fractional iteration is well defined, as you have nicely behaved flow from one stationary point to the other. Of course, this operation has to satisfy rules, such as $f^{1/2}(f^{1/2}(x))=f(x)$ and so on. I don't know enough about the topic to state it with certainty, but I think there is no generally accepted rule for computing it for an arbitrary function, especially not in closed form (I once tried to compute it numerically for a few noninteger "powers", with mixed results). I do know that the question may be rewritten in terms of Schröder's equation, which looks more like a normal equation than asking "what happens if we apply a function noninteger number of times".

This is also somewhat related to the concept of fractional differentiation.