Note. The answer below may be excessive compared to what OP is asking.
For a quick answer, read the definition of $(x_n)$, jump directly to the proof of proposition, and then read only the first 3 steps.
Let $a_1 \in (0, 1)$ and define $x_n = 1/a_n$. Then $x_n$ solves the following recurrence relation
$$ x_{n+1} = x_n + 1 + \frac{1}{x_n} + \frac{1}{x_n(x_n - 1)}. \tag{1}$$
Using this we progressively reveal the asymptotic behavior of $(x_n)$. More precisely, our goal is to prove the following statement.
Proposition. Let $(x_n)$ be defined by $\text{(1)}$, i.e. $x_1 > 1$ and $x_{n+1} = f(x_n)$ for $f(x) = \frac{x^2}{x-1}$. Then there exists a function function $C : (1, \infty) \to \mathbb{R}$ such that
$$ x_n = n + \log n + C(x_1) + \mathcal{O}\left(\frac{\log n}{n}\right) \quad \text{as} \quad n\to\infty. $$
Here, the implicit constant of the asymptotic notation may depend on $x_1$. Moreover, $C$ solves the functional equation $C(f(x)) = C(x) + 1$.
We defer the proof to the end and analyze the asymptotic behavior of OP's limit first. Plugging the asymptotic expansion of $x_n$, we find that
$$ r_n := \frac{n(1-n a_n)}{\log n} = \frac{n(x_n - n)}{x_n \log n} = 1 + \frac{C(x_1)}{\log n} + \mathcal{O}\left(\frac{\log n}{n}\right). $$
This tells that, not only that $r_n \to 1$ as $n\to\infty$, but also that the convergence is extremely slow due to the term $C/\log n$.
For instance, $f^{\circ 94}(2) \approx 100.37$ tells that $C(100) \approx C(2) + 94$. Indeed, a numerical simulation using $n = 10^6$ shows that
\begin{align*}
x_1 = 2 &\quad \Rightarrow \quad (r_n - 1)\log n \approx 0.767795, \\
x_1 = 100 &\quad \Rightarrow \quad (r_n - 1)\log n \approx 94.3883,
\end{align*}
which loosely matches the prediction above.
Proof of Proposition.
Step 1. Since $x_{n+1} \geq x_n + 1$, it follows that $x_n \geq n + \mathcal{O}(1)$. In particular, $x_n \to \infty$ as $n\to\infty$.
Step 2. Since $x_{n+1} - x_n \to 1$, we have $\frac{x_n}{n} \to 1$ by Stolz-Cesaro theorem.
Step 3. Using the previous step, we find that
$$ \frac{x_{n+1} - x_n - 1}{\log(n+1) - \log n} = \frac{1}{(x_n - 1)\log\left(1+\frac{1}{n}\right)} \xrightarrow[n\to\infty]{} 1 $$
So, again by Stolz-Cesaro theorem, we have $x_{n+1} = n + (1+o(1))\log n$. This is already enough to conclude that OP's limit is $1$.
Step 4. By the previous step, we find that $ x_{n+1} - x_n
= 1 + \frac{1}{n} + \mathcal{O}\left(\frac{\log n}{n^2}\right)$. Using this, define $C$ by the following convergent series
$$ C(x_1) = x_1 - 1 + \sum_{n=1}^{\infty} \underbrace{ \left( x_{n+1} - x_n - 1 - \log\left(1+\frac{1}{n}\right) \right) }_{=\mathcal{O}(\log n/n^2)}. $$
Splitting the sum for $n < N$ and $n \geq N$ and using the estimate $\sum_{n\geq N}\frac{\log n}{n^2} = \mathcal{O}\left(\frac{\log n}{n}\right)$,
$$ C(x_1) = x_N - N - \log N + \mathcal{O}\left(\frac{\log N}{N}\right), $$
which confirms the first assertion of the proposition.
Once this is established, then the second assertion easily follows by interpreting $x_{n+1}$ as the $n$-th term of the sequence that solves $\text{(1)}$ with the initial value $f(x_1)$. Hence comparing both
$$ x_{n+1} = n+1 + \log(n+1) + C(x_1) + o(1) $$
and
$$ x_{n+1} = n + \log n + C(f(x_1)) + o(1) $$
the second assertion follows. ////
Best Answer
Assume that $a_n> 0$ for every positive integer $n$; otherwise the limit may not exist. Set $z_n:=n^x\,a_n$ like Martin R recommended. Thus, $$n^x\,\left(\prod_{i=1}^n\,a_i\right)^{\frac{1}{n}}=\frac{n^x}{\left(\prod\limits_{i=1}^n\,i^x\right)^{\frac1n}}\,\left(\prod\limits_{i=1}^n\,z_i\right)^{\frac1n}=\left(\frac{n^n}{n!}\right)^{\frac{x}{n}}\,\left(\prod_{i=1}^n\,z_i\right)^{\frac1n}\,.$$ Now, since $\displaystyle\lim_{n\to\infty}\,z_n=a$, we have $\displaystyle\lim_{n\to\infty}\,\left(\prod_{i=1}^n\,z_i\right)^{\frac1n}=a$ (see Martin R's link in the comments above). Furthermore, Stirling's approximation $n!\approx \sqrt{2\pi n}\left(\frac{n}{\text{e}}\right)^n$ implies that $$\lim_{n\to\infty}\,\left(\frac{n^n}{n!}\right)^{\frac{x}{n}}=\lim_{n\to\infty}\,\left(\frac{\text{e}^n}{\sqrt{2\pi n}}\right)^{\frac{x}{n}}=\exp(x)\,.$$ Consequently, $$\lim_{n\to\infty}\,n^x\,\left(\prod_{i=1}^n\,a_i\right)^{\frac{1}{n}}=a\,\exp(x)\,.$$