A connected Lie group \(G\) is called spacious if there exists an open subset \(U \subseteq G\) such that \(U^n \cap U^{n + 1} = \emptyset\) for all \(n \in N\). This property is closely related to the behaviour of the exponential function \(\text{exp} : g \to G\) because, according to a result of Jaworski, \(G\) is spacious if and only if \(\text{exp }g\) is not dense in \(G\) [\textit{W. Jaworski}, The density of the image of the exponential function and spacious locally compact groups (submitted)]. Jaworski characterizes the spacious semisimple groups as those where the minimal parabolic subgroups are disconnected. Now let us call \(G\) completely spacious if \(G \setminus \text{exp }g\) contains an open subsemigroup of \(G\) and note that this obviously implies that \(G\) is spacious. If \(R\) is the radical of \(G\), then \(G\) is completely spacious if and only if the semisimple group \(G/R\) is completely spacious. The main result of this paper is a characterization of the semisimple completely spacious groups in the same spirit as Jaworski's result as those where the minimal parabolic subgroups have infinitely many connected components. This conditions turns out to be equivalent to the existence of a Cartan subgroup with infinitely many connected components.