A variant of $\ell^2$-cochains Suppose $X$ is an infinite countable CW complex which satisfies the following property: for all $k$-cells $e$, the  number of $(k+1)$-cells incident to $e$ is at most $c_k$, where the latter is some number that depends on $k$. Let $X_k$ be the set of $k$-cells.
Let $\ell^2_k(X)$ be the set of functions $a : X_k \to \Bbb R$, 
 such that the series
$$
\sum_{e \in X_k} a(e)^2
$$
converges (this implicitly makes use of the counting measure on $X_k$).  Then the incidence bound assumption implies that
coboundary operator
$$
\delta: \ell^2_k(X) \to \ell^2_{k+1}(X)
$$
is defined (this uses the same formula that arises when defining the cellular cochain complex of $X$).
When $\dim X =1$ this construction was introduced by Dodziuk and Kendall in
Dodziuk, J.(1-CUNYG); Kendall, W. S.(4-STRA)
Combinatorial Laplacians and isoperimetric inequality. From local times to global geometry, control and physics (Coventry, 1984/85), 68–74, 
Pitman Res. Notes Math. Ser., 150, Longman Sci. Tech., Harlow, 1986. 
Questions


*

*Has this construction been investigated in the generality described above?

*How is the cohomology of this complex related to the usual cellular cohomology of $X$?

*Is there a set of reasonable conditions on $X$ which guarantee that this cohomology is finite dimensional?

*How does the above relate to other notions of $L^2$-cohomology?
 A: I'm not sure if this should be a comment rather than an answer, since I'm not attempting to answer the hard questions (2) and (3).  But it's too long to be a comment.  That said:
First of all, I just want to clarify that presumably you want incidence to be bounded with multiplicity, that is, the coefficients of the boundary map to also be bounded.
For (4), take a look at this technical report by Attie and Block:
https://pdfs.semanticscholar.org/cfd5/297840f262e255f86777e112757dbffdf9e5.pdf
Theorem 4 is a de Rham theorem for $L^p$ cohomology -- for a manifold with bounded geometry, $L^p$ cohomology is equal to simplicial $L^p$ cohomology with real coefficients, on any given bounded triangulation.  (I suppose you can also define a "piecewise de Rham" cohomology and then the isomorphism holds for any simplicial complex, in standard and $L^p$ flavors.)
Now if you have a manifold with a nice enough CW structure, then you could replace each cell with a bounded number of simplices up to homotopy equivalence, and you would have an isomorphism from your theory to the "usual" de Rham version.  The pullback to the universal cover of a CW structure on a compact manifold is of course nice enough, but a CW complex of bounded incidence is not.  E.g. take an $\mathbb{R}$ and at the $n$th lattice point attach an $S^2$ which in turn has a 4-cell attached to it via a map with Hopf invariant $n$.  This has bounded incidence, since only the 1-cells are incident to anything at all, but an infinite number of homotopy types of attaching maps and so a simplicial complex version would have to get more and more complicated as you go further out.
I don't know of anyone studying the $L^2$ version in this level of generality.  I studied the $L^\infty$ analogue at some length in my thesis (see sections 5 and 6 of https://arxiv.org/abs/1410.3368), but I was only concerned with universal covers of compact spaces, which avoids a lot of technical issues.
