It has been previously proved that the number of integers for which the number of digits in each of the base-\(\theta\) and base-\(\psi\) expansions lies below a fixed bound is finite if and only if 1, \(\theta\) and \(\psi\) are \(\mathbb{Q}\)-linearly independent. In this talk, we prove various results which lead us to showing that the number of integers which have finitely many digits in the Ostrowski expansions with respect to irrational numbers \(\alpha = \frac{1 + \sqrt{5}}{2}\) and \(\beta = \frac{3 + \sqrt{13}}{2}\) is bounded. This is joint work with Alan Haynes.