It is a long-standing question whether an arbitrary variety is desingularized by finitely many normalized Nash blowups. We consider this question in the case of a toric variety. We interpret the normalized Nash blowup in polyhedral terms, show how continued fractions can be used to give an affirmative answer for a toric surface, and report on a computer investigation in which over a thousand 3- and 4-dimensional toric varieties were successfully resolved.