The concept of dimension in simple games was introduced by Taylor and Zwicker (1993) as a measure of the remoteness of a given game from a weighted game. They demonstrated that the dimension of a simple game can grow exponentially in the number of players. However, the problem of worst-case growth of the dimension in complete games was left open. Freixas and Puente (2008) showed that complete games of arbitrary dimension exist and, in particular, their examples demonstrate that the worst-case growth of dimension in complete games is at least linear. In this paper, using a novel technique of Kurz and Napel (2015), we demonstrate that the growth of dimension in complete games can also be exponential in the number of players.