In the definition of an adic space, usually there is a presheaf defined by first saying what it is on a particular basis of the topology of the underlying space, the so called rational subsets. One then extends this to arbitrary opens by taking the limit over all the rational subsets inside the given open. However, in all the references I looked into so far, it is then quickly pointed out that this is in general not a sheaf, followed by a list of particular cases in which it is one. 


In Algebraic Geometry, there are many occasions where some construction does not give a sheaf and one simply forces it to be one by saying "Sheafify!" So the Question is:


-Why doesn't one sheafify the structure presheaf of an adic space? Is there no sheafification functor in this case? If not, what goes wrong?



why don't (can't?) we sheafify the structure presheaf of an adic space
There is actually a sheafification process used in classical construction of rigid geometry, which is used to pass from the weak G-topology to the strong one (see section 9.2.2 of Bosch, Güntzer, Remmert's Non-Archimedean Analysis, for instance). 


As to why people don't use it in the case of non-sheafy spaces, I would say that if depends on what you want to do. If it's enough for you to know the stalks of the structure sheaf, then fine. But, as Dylan Wilson pointed it out, you are likely not to end up with the global sections you expect. For an explicit example, see Theorem 3.15 in Mihara's paper "On Tate Acyclicity and Uniformity of Berkovich Spectra and Adic Spectra" (<http://arxiv.org/abs/1403.7856>).


