The real function $f$ such that $\log \cdots \log (f)$ is strictly convex on its domain for any number of $\log$'s

After realizing the flaw in my previous, affirmative, answer: No such function exists.

If $f$ has a nonempty domain at every point, then it grows arbitrarily large, but since it is continuous, after sufficiently many iterations of log its range will always be $(-\infty,\infty)$. WLOG, take $f$ to be such a function (if any valid $f$ exists, all of its logarithms are valid too). It is strictly monotonically increasing (as it is convex and not bounded below), so we let $c_0<c_1<c_2<\ldots$ be the unique points at which $f$ attains the values $0,1,e,e^e,e^{e^e},\ldots$.

Now consider $g=\log(f)$. We have $g(c_1)=0, g(c_2)=1$, and that $g$ is convex and monotonically increasing. We also have $\lim_{x\to c_0}g(x)=-\infty$. But then consider the line from $(x,g(x))$ to $(c_2,1)$; as $x$ approaches $c_0$ from the positive direction, this line will drop below the point $(c_1,0)$, violating convexity.

It is possible for $f$ and all logs applied to it to be convex on all positive values, as outlined in my other answer, but not over its whole domain.