When one thinks in a less-mathematical, more physical way about this problem, it becomes obvious that there are significant problems immediately. I consider the problem as set up by colonelmustard - constant charge density rho in 3 dimensions, and we desire to find the potential (or the electric field which obeys Maxwell; either will do).
Consider an origin O again, and let points we consider be at distance away from this point R (in spherical co-ordinates). Now, consider the spherical shells again. They will be distance R away from the origin, with charge per metre rho*C*R*R total, and be subject to the inverse square law so that the potential phi at O will see a contribution of D*rho*(R*R/R*R) from this shell, where C and D are some (probably dimensioned) constants I don't care about.
Uh-oh. This is a finite contribution from a shell at distance R away - a contribution of rho*D (it doesn't matter what it is - it matters what it isn't - it isn't a function of R). I see immediately that if I add up the value of an infinite number of shells that I'm going to get an infinite potential at every point. I conclude I oughtn't proceed until I've thought a bit harder.
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See also: Olbers' Paradox. This is identical, mathematically, until the resolution of the paradox.