On eventually always hitting points

We consider dynamical systems $(X,T,\mu)$ which have exponential decay of correlations for either H\"older continuous functions or functions of bounded variation. Given a sequence of balls $(B_n)_{n=1}^\infty$, we give sufficient conditions for the set of eventually always hitting points to be of full measure. This is the set of points $x$ such that for all large enough $m$, there is a $k < m$ with $T^k (x) \in B_m$. We also give an asymptotic estimate as $m \to \infty$ on the number of $k < m$ with $T^k (x) \in B_m$. As an application, we prove for almost every point $x$ an asymptotic estimate on the number of $k \leq m$ such that $a_k \geq m^t$, where $t \in (0,1)$ and $a_k$ are the continued fraction coefficients of $x$.