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
EDIT Corrected a couple of inaccuracies and mistakes, added some references.
For the sake of clarity, let me work with non-negatively graded cochain complexes, and analyze the case of a left exact covariant functor, to prove that its right derived functor is a universal (covariant) cohomological $\delta$-functor.
Let $\mathcal A$ and $\mathcal B$ be two abelian categories, and let $F : \mathcal A \to \mathcal B$ be a left exact functor. Let me denote by $\gamma_\mathcal{A}$ the localization functor $\mathrm{Ch}_+(\mathcal{A}) \to \mathrm{D}_+(\mathcal{A})$. Let $\tilde S_\mathcal{A}$ denote the full subcategory of $\mathrm{Ch}_+(\mathcal A)$ consisting of discrete complexes (i.e. of complexes whose differentials are all zeroes). Let $S_\mathcal{A}$ denote its essential image in the derived category.
First of all, observe that a $\delta$-functor is a particular case of what MacLane calls "connected sequences" (connected sequence is a $\delta$-functor iff the long sequence it induces is exact). Now, defining a connected sequence is equivalent to define a functor $S_\mathcal{A} \to \mathcal{B}$ (see Proposition XII.8.1 and Proposition XII.8.2 in [1]). Part of the equivalence works as follows: given any $T : S_\mathcal{A} \to \mathcal{B}$, define $T^n(A) := T(A[n])$, and check that, given any exact sequence $0\to A\to B\to C\to 0$, the image of the class corresponding to it in
$$S_\mathcal{A}(C[n],A[n+1]) \simeq \mathrm{D}_+(\mathcal A) (C[n], A[n+1]) \simeq \mathrm{Ext}^1_{\mathcal A}(C,A)$$
under $T$ gives you a morphism $\delta^n : T^n C \to T^{n+1} A$ giving $\{T^n\}$ the structure of a connected sequence. You can find the details of this construction in [1].
We have that
$$\mathbb{R}F := \mathrm{Lan}_{\gamma_\mathcal{A}(-[0])} (\gamma_\mathcal{B} F(-)[0]).$$
Now, we can postcompose the bottom corners of the square defining $\mathbb{R}F$, along the functors $$\gamma_{\mathcal A} \circ \bigoplus_{n\geq 0} H^n : \mathrm{D}_+(\mathcal{A}) \to S_\mathcal{A}$$ and
$$H^0 : \mathrm{D}_+(\mathcal{B}) \to \mathcal{B}$$ as follows
$$
\require{AMScd}
\begin{CD}
\mathcal{A} @>{F}>> \mathcal{B}\\
@VVV @VVV \\
\mathrm{D}_+(\mathcal{A}) @>{\mathbb{R}F}>> \mathrm{D}_+(\mathcal{B})\\
@VVV @VVV \\
S_\mathcal{A}@. \mathcal{B}
\end{CD}
$$
and further Kan extend, obtaining the connected sequence $RF : S_\mathcal{A} \to \mathcal{B}$ as
$$\mathrm{Lan}_{\gamma_\mathcal{A} \circ \oplus_n H^n \circ (-)[0]}(H^0 \circ (-)[0] \circ F) \simeq \\ \simeq \mathrm{Lan}_{\gamma_\mathcal{A} (-[0])}(F).$$
Now, given any other connected sequence (in particular, any $\delta$-functor) $T : S_\mathcal{A} \to \mathcal{B}$, by definition of left Kan extension we have
$$\mathrm{Nat}(RF,T) \simeq \\ \simeq \mathrm{Nat}(F, T \circ \gamma_{\mathrm{A}} (-[0])) \simeq \\ \simeq \mathrm{Nat}(F,T^0)$$
showing that the "universality" of $RF$ works not only for $\delta$-functors, but in general for connected sequences.
The fact that for $F$ left exact and $\mathcal A$ with enough injectives one has that $RF$ is not only a connected sequence, but really a $\delta$-functor, follows from XII.8.3, 4 and 5 in [1].
[1] MacLane - Homology
Best Answer
Some years ago I taught some homotopy coherence theory mentioning quasi-categories, etc in a Masters course on cohomology in Ottawa, and then used the material from that in several graduate level mini-courses at conferences. The course used crossed modules etc. and was not as detailed on Higher Algebra as perhaps you are meaning. I wrote up the notes and then extended them adding in material for the mini-courses as I went and filling in background. Anyone is welcome to have a look and to use the notes . The notes have got very long in their current form but I have another copy available on my n-lab page at: http://ncatlab.org/timporter/files/menagerie11.pdf
Other forms of the notes are available although none fits exactly the original questioner's requirements. They have been used on several occasions with moderate success. Have a look at the other notes that are linked to from my n-Lab page: http://ncatlab.org/timporter/show/HomePage as they may have some material that is usable.
That deals with available material. Prior to that(<2006) we taught numerous masters level courses at Bangor (when there was still a Maths Dept there) which incorporated material on crossed modules, crossed complexes, their infinity groupoid interpretations, etc. and their use in non-Abelian cohomology and homological algebra. The links with various variants of infinity category theory were explored in those courses.