It is a known fact [1] that, for every $c\in (1,\infty]$ it is possible to find an alphabet $\mathcal{A}$ and a word $w\in \mathcal{A}^\omega$ such that $w$ has critical exponent $c$. It looks natural to me to define what I would call the critical exponent function in two steps as follows.

Step 1:

The $n$-critical exponent function $\kappa_n$: \begin{equation} x\in[0,1]\;\longmapsto \;\kappa_n(x)=\frac{1}{\mathit{c.e.}(x)} \end{equation} where $\mathit{c.e.}(x)$ is the critical exponent of the infinite $n$-base expansion of $x$ (and it is intended $\frac{1}{\infty}=0$).

The range of $\kappa_n$ is $[0,\frac{4}{7}]$ for $n=3$, $[0,\frac{5}{7}]$ for $n=4$ and $\left[0,\frac{n-1}{n}\right]$ when $n=2$ or $n\ge 5$. This is due to a result by Rao [2] covering the last cases of a general conjecture by Dejean on repetition thresholds for finite alphabets [3].

Step 2:

The critical exponent function $\kappa$: \begin{equation} \kappa: x\in[0,1]\;\longmapsto\; \sup_{n\in\mathbb{N}}{\kappa_n(x)}\in[0,1] \end{equation}

It is easily seen that $\kappa(x)=0$ for every $x\in [0,1]$ such that its expansion in every base admits densities of the digits – in particular $\kappa$ vanishes on absolutely normal real numbers. Therefore, $\kappa$ is Lebesgue-measurable and $\int_0^1 \kappa(x)dx=0$. Notice also that $\kappa$ is not affected by the possible ambiguity in the expansion of some rational points (the critical exponent is $\infty$ for both the possible expansions).

Looks like $\kappa$ has several unusual properties (it recalls loosely Conway's base-13 function). Almost every question about it I can think of seems non-trivial. I propose three of them.

Q1: Which Baire class (if any) does this function belong to?

Q2: Does $\kappa$ have fixed and/or periodic points (apart from the trivial fixed point 0)?

Q3: Does $\kappa$ attain the value 1?

[1] Krieger, D., & Shallit, J. (2007). Every real number greater than 1 is a critical exponent. Theoretical computer science, 381(1-3), 177-182.

[2] Rao, M. (2011). Last cases of Dejean's conjecture. Theoretical Computer Science, 412(27), 3010-3018.

[3] Dejean, F. (1972). Sur un théorème de Thue. Journal of Combinatorial Theory, Series A, 13(1), 90-99.