Let $L(X) = e^{\sqrt{(\ln X)(\ln \ln X)}}$. We claim that $L(X)$ is subexponential.

\[\begin{align*} e^{\sqrt{(\ln X)(\ln \ln X)}} &> e^{\sqrt{(\ln \ln X)(\ln \ln X)}} \\ &= e^{\ln \ln X} \\ &= \ln X. \end{align*}\] \[\begin{align*} e^{\sqrt{(\ln X)(\ln \ln X)}} &< e^{\sqrt{(ln X)(\ln X)}} \\ &= e^{\ln X} \\ &= X. \end{align*}\]