Is complete genetic linkage possible?
Complete genetic linkage does occur in some cases. 

Most of the Y chromosome (present as a single copy in males of many species) does not recombine with the X chromosome. Any genes in this non-recombining region, the "male specific Y" (which is 95% of the Y), will be "completely linked", in other words, they cannot be broken apart by recombination and are inherited as a single unit.



Non-recombining region of the Y marked as NRY from this paper.

The closer two genes are, the higher the general tendency for them to be co-inherited. When co-inheritance is non-random the genes are in linkage disequilibrium. The genetic linkage between genes is measured in centimorgans.

I'm not aware of any completely linked genes in recombining parts of the genome. By definition there is some distance between two linked genes (otherwise it's one pleiotropic gene). Therefore, if recombination is possible, then it is possible for them to become unlinked at some frequency, even if that frequency is extremely small (I could be wrong though, but scholar searches have given me nothing). There are, however, cases of very tight linkage between genes (1, 2, 3).

Note that factors other than genetic linkage can affect linkage disequilibrium.