  Let $[a_1(x),a_2(x), a_3(x),\cdots]$ denote the continued fraction expansion
of a real number $x \in [0,1)$. This paper is concerned with certain
exceptional sets of the Borel-Bernstein Theorem on the growth rate of
$\{a_n(x)\}_{n\geq1}$. As a main result, the Hausdorff dimension of the set \[
E_{\sup}(\psi)=\left\{x\in[0,1):\ \limsup\limits_{n\to\infty}\frac{\log
a_n(x)}{\psi(n)}=1\right\} \] is determined, where
$\psi:\mathbb{N}\rightarrow\mathbb{R}^+$ tends to infinity as $n\to\infty$.
