This is something I was thinking about, so I'm going to post it as a question and post my own answer. I hope that anyone who wants will comment, correct me if I'm wrong, and add their own knowledge and thoughts.
Suppose $f:[a,b]\to \mathbb{R}$ is a positive, continuous function. The $L^p$ norms of $f$ are the analogs to the weighted power means of a group of positive numbers $a_1, \dotsc, a_n$. (Perhaps it would be more precise to say that the expressions $\left(\int_a^b\frac{f^p}{b-a}dx\right)^{1/p}$ are the analogs.) The power means are given by $$M_p(a_1, \dotsc, a_n)=(w_1a_1^p + \dotsb + w_na_n^p)^{1/p},$$
where $w_i$ such that $\sum w_i =1$ are any weights we like (the simplest case is when $w_i=1/n$ for all $i$). It is well-known (see here) that the power means $M_p$ tend to the geometric mean as $p\to 0$. What does $$\left( \frac{\int_a^b f(x)^p}{(b-a)}\right)^{1/p}$$
tend to as $p\to 0$?
Since you normalized the measure space to have total measure $1$, Jensen's inequality immediately implies the monotonicity of $L^p$ norm (quasinorm for $p<1$) with respect to $p$. Namely, for all $p>0$ and all $r>1$ we have (using the convexity of $t\to t^r$ on $[0,\infty)$)
$$ \left(\int_X |f|^{p}\right)^{r} \le \int_X (|f|^p)^r $$ hence $\|f\|_p\le \|f\|_{pr}$.
You can also use Hölder's inequality to the same purpose.
Thus, the limit $\lim_{p\to 0}\|f\|_p$ exists under the rather weak assumption that some $L^p$ norm of $f$ is finite. As you noted, it is equal to $\exp\left(\int_X \log|f| \right)$. Continuity and the particular form of measure space are irrelevant.