A recent blog post by Jeff Elly brought to my attention “NAJ Economics“, an overlay journal devoted to Economics that has been in operation for about eight years, although at pretty low volume. “NAJ” stands for “Not a Journal” (or the geek-wannabe GNU-like “NAJ aint a journal”), and the idea is that the set of (rather distinguished) editors choose papers that they like on the web, peer-review them, and publish links to the “reviewed” papers. The way it works is that the editors pick what they want to review: you can not submit your paper to NAJ, nor can you ask them not to review your paper once you’ve put it openly on the web. The idea is that this gives a peer-reviewed publication: the author takes care of publication on the web, and NAJ provides the peer-review. (I started thinking about “NAJ AGT” which could handle publication more elegantly by relying on the arXiv. But then, it doesn’t seem that “NAJ economics” is a success story, so maybe not.)
At the same time, Daniel Lemire posts a 1987 “EWD” by Dijkstra going against the whole notion of trusting peer review too much. Indeed, Dijkstra rarely published his work in the usual sense but rather mailed out photocopies of his hand-written writings, termed EWDs, to colleagues. While my sympathies (like those of Lemire) are with this mode of publication, I’m afraid that few non-Turing-Award-winners will get their work noticed this way, so some mechanism for allocation of attention is still needed.
Turing's Invisible Hand 
Perhaps a better solution would be to keep the current framework where submitting a paper to a journal guarantees a review within some timeframe. When a paper gets accepted, however, instead of it being printed it would just be added to a publicly available list of accepted papers with a link to arXiv. The sole purpose of the journal would then be to act as a stamp of recognition, such that the authors can add the paper to their publication lists and others will get a rough estimate of the scope and quality of the paper.
Even though, ideally, we should do research for the greater good, I think people tend to, selfishly, focus on improving their cv and thereby their chances of getting a job. From that point of view the stamp of recognition itself is at least as important as the dissemination, which I guess is the main reason journals are still around. Remember that the vast majority of papers only get cited very few times, if at all. In the long run the main contribution of such papers are to the authors’ cv. This is of course undesirable, but probably very hard to fix.
Being cynical, your list of publication somehow corresponds to your rating in, for instance, chess. If everyone was assigned some number representing their skill level, and you were hired based on your rating, there would probably be fewer insignificant papers. Right now, however, your rating is basically the frequency at which you can publish papers, which means that an awful lot of papers are being produced.
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