Is there a sense in which $\mathbb{R}$ does have 'holes'?

Once the rational numbers $\mathbb Q$ are extended to a Dedekind-complete field, there is no room to add any further numbers, so long as one is working with an Archimedean number system. The intuitive picture that "there are no more holes" is perhaps helpful in grasping this mathematical fact, but it can be detrimental if extended further to some kind of notion of impossibility of even larger (and "more closely packed") number systems. Indeed such systems do exist; one of them that's particularly useful in analysis is the hyperreal number system. In this sense $\mathbb R$ does have holes.