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In the paper we provide measure estimates for the set of numbers whosesequence of products of continued fraction partial quotients $M_n = a_1 \ldotsa_n$ has exponential growth with rate close to the one predicted byKhintchine's theorem, i.e. for which  \begin{equation*}  e^{(\kappa - T)n} \leqslant M_n \leqslant e^{(\kappa + T)n}  \end{equation*} for a fixed $T > 0$ and all $n$ greater than some fixedinteger $N$, where $e^\kappa = 2.685\ldots$ is the Khintchine constant.Choosing $N$ large enough the measure can be made arbitrarily close to full,for any given $T$. The bounds are not of asymptotic nature, but explicit interms of the parameters involved. In the proof we compile several known resultof large deviations theory, employing the cumulant method in particular. Wealso discuss the numerical values of the quantities involved.

A quantitative version of the theorem on Khintchine's constant