What does a double parenthesis in a quoted value ─ like e.g. 157(3)(3) ─ mean? Sometimes, in journal articles, the author writes a number followed by two parenthesis. For example, $157(3)(3)$, where it seems that the first parenthesis shows uncertainty in the last digit (here, $7$), but I am not sure what the second parenthesis shows.
 A: As noted by rob, this will very often be a simultaneous reporting of both statistical and systematic uncertainties in the quoted value.
However, I would be extremely surprised if there are subfields where this notation has become standard enough that it is considered acceptable to use it without specifying clearly which uncertainty is which. Thus, for any paper that uses double-parenthesis notation to report multiple uncertainties on the same value, I would expect a clear and unambiguous definition of what the notation represents. This will generally be within half a page or so from the first usage of the notation.
For the example you mentioned (arXiv:1005.3508), this is indeed explicitly laid out:



This should be enough cause for concern: rob's gut-feeling interpretation (statistical then systematic) is not what's being reported in the paper. My inference from that conflict is that any paper that uses this notation without clearly defining it is at fault for sloppy writing and should be taken to task on that. (However, I expect that this isn't the case - if you go back to the examples you've found, I expect that they will all provide such suitable definitions.)
A: Generally when two uncertainties are reported, one is the "statistical" uncertainty and the other is the "systematic" uncertainty.
The statistical uncertainty arises in experiments where it's only possible to observe a small number of events. (Or a large number of events but a very small observable --- I've done some experiments where we've collected $10^{18}$ events in order to measure a part-per-billion asymmetry.)
The systematic uncertainty arises in experiments where the apparatus has some bias that must be accounted for. For example, in a radiation-counting experiment, it's usually impossible to build an array of detectors that will interact with every particle that you'd like to detect: some particles will pass through a detector without depositing energy, and some will pass through gaps in the array, and some will arrive at the same time as another and register as one big event rather than two small events. You have to correct for this, and the correction introduces some uncertainty of its own.
When separate reporting of systematic and statistical errors began to appear in the literature, people would be explicit about it, and you would see things like
$$
157\pm3_\text{stat}\pm3_\text{syst}
$$ 
But perhaps now that the novelty is wearing off, people are starting to get sloppy.
