  We consider level-2 large deviations for the one-sided countable full shift
without assuming the existence of Bowen's Gibbs state. To deal with non-compact
closed sets, we provide a sufficient condition in terms of inducing which
ensures the exponential tightness of a sequence of Borel probability measures
constructed from periodic configurations. Under this condition we establish the
level-2 Large Deviation Principle. We apply our results to the continued
fraction expansion of real numbers in $[0,1)$ generated by the R\'enyi map, and
obtain the level-2 Large Deviation Principle, as well as a weighted
equidistribution of a set of quadratic irrationals to equilibrium states of the
R\'enyi map.
