  We introduce the notion of weakly (strongly) infinite real rank for unital
$C^{\ast}$-algebras. It is shown that a compact space $X$ is weakly (strongly)
infine-dimensional if and only if $C(X)$ has weakly (strongly) infinite real
rank. Some other properties of this concept are also investigated. In
particular, we show that the group $C^{\ast}$-algebra $C^{\ast}({\mathbb
F}_{\infty})$ of the free group on countable number of generators has strongly
infinite real rank.
