Sorry for the late reply. I am mostly not in front of the computer/cell-phone during the week-end (I am already spending more than 70 hours in front of them during the week :p )

Double sorry for having ignored the ALICE comparison. I just forgot to comment... thanks for insisting. Just to start with, what is said below probably holds for all LHC experiments. Then, just a disclaimer. I want to recall that I am a theorist, so that I will try to answer as good as I can. But this level of technicality is not directly connected to what I do.

The triggers are the key. Not all collisions will be recorded and decisions are made so that only interesting collisions will be recorded and that should satisfy some electronic constraints. The challenges were, at the time the experiment was designed, to be able to have good triggers than guarantee that all interesting events would be recorded, and that the data acquisition system was good enough to follow the rhythm. I think the the challenges have been successfully achieved.

Note that in this chain, you also have Tier 2 and Tier 3 machines so that the analysis work is done on the lower machines that can request some part of the data with some filters.

I hope all of this helps. Do not hesitate to continue the discussion ^^