As far as being larger than infinitely-sized objects or spaces goes, one must analyze the context of the feat in question to determine if it truly qualifies for Higher-Dimensional Existence. In terms of volume (Or, more generally, measure), the only way to be truly bigger than an object of infinite size is to have a non-zero size in a space of more dimensions than the object in question. However, portrayals of more expansive realms containing infinitely large things within themselves are not necessarily indicative of such.
A good construction to exemplify this is the topological space known as the long line. In essence, it is a space obtained by taking an uncountably infinite number of line segments and “gluing” them together end-to-end, and so it is in some sense much longer than the real line, which is comprised of only a countably infinite number of such line segments. Nevertheless, they are both 1-dimensional spaces.
The long line itself can also be generalized into 2-dimensional and 3-dimensional analogues, and as such the same principle holds for higher dimensions as well.
