Summary: We consider dynamical systems \((X,T,\mu)\) which have exponential decay of correlations for either Hölder 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 \(m\rightarrow\infty\) 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 \(k \le m\) such that \(a_k\ge m^t\), where \(t\in (0,1)\) and \(a_k\) are the continued fraction coefficients of \(x\).