How to evaluate $\frac{1}{a_{1}+\frac{1}{a_{2}+\frac{1}{a_{3}+\frac{1}{a_{4}+\frac{1}{\ddots}}}}}$, where $a_{j}=\frac{1}{1}+\frac{1}{2}+\frac{1}{3}+\dots+\frac{1}{j}$?
That is,
How to evaluate
$\frac{1}{\left ( \frac{1}{1} \right )+\frac{1}{\left ( \frac{3}{2} \right)+\frac{1}{\left ( \frac{11}{6} \right )+\frac{1}{\left ( \frac{25}{12}\right )+\frac{1}{\ddots}}}}}$?
The value of the expression is $0.6606\dots$, but what is its exact value in a simple form, using fractions, radicals, logarithms, etc.?