I don't think the existence of the dual "injective" model structure merits an "of course," since its generators are much less obvious to construct. However, it turns out that injective model structures actually exist in more generality than projective ones, for instance they exist for most categories of sheaves. I believe this was originally proven by Joyal, but it was put in an abstract context by Hovey and Gillespie.
The basic idea is that model structures on Ch(A) correspond to well-behaved "cotorsion pairs" on A itself. The projective model structure comes from the (projective objects, all objects) cotorsion pair (which is well-behaved for R-modules, but not for sheaves), and the injective one comes from (all objects, injective objects). There is also e.g. a flat model structure coming from (flat objects, cotorsion objects) which is monoidal and thus useful for deriving tensor products. A good introduction, which I believe has references to most of the literature, is Hovey's paper Cotorsion pairs and model categories.
The process can certainly be iterated as explained by Marc (see also Weibel, Homological Algebra, 1.2.5. Moreover cf. 1.2.3, 2.2.2 for the fact that the category of chain complexes over an abelian category with enough projectives is again an abelian category with enough projectives).
However, it seems to me that it isn't often used. A reason might be, that in many (most ?) cases one isn't interested in a double complex (or higher dimensional analogs) itself but in the (co)homology of of its total complex (the definition of hyperderived functors in your question is an example for this point of view).
However, in this case there is no need to jump into a higher dimension to define a projective resolution. For, there is an alternative definition for the projective resolution of a chain complex that is - in my opinion - much more elegant and easier to work with than with Cartan-Eilenberg's definition:
A projective resolution of the chain complex $C$ is a complex $P$ of projectives together with a quasi-isomorphism $f: P \to C$ (i.e. $f$ is a chain map such that $H_n(f): H_n(P) \to H_n(C)$ is an isomorphism for all $n$).
Note that such a $P$ is in general no projective object in the category of chain complexes, but it yields the same hyperderived functors, hypercohomology spectral sequences, etc. For a textbook reference of this definition see for example
- McCleary, A User's Guide to Spectral Sequences (before Theorem 12.12)
- Benson, Representations and Cohomology I, Definition 2.7.4
Best Answer
Among the standard examples of abelian categories without enough projectives, there are
No abelian category where the functors of infinite product are not exact can have enough projectives. In Grothendieck categories (i.e. abelian categories with exact functors of small filtered colimits and a set of generators) there are always enough injectives, but may be not enough projectives.
Among the standard examples of abelian categories with enough projectives, there are