This is a common suggestion and it is used more and more. The problem is the data rate: with LHC you have a flood of interesting events to record in a very cont

This is a common suggestion and it is used more and more. The problem is the data rate: with LHC you have a flood of interesting events to record in a very controlled environment. On the other hand, super-high-energy cosmic rays are very rare and frequently you do not know many of the initial parameters necessary to interpret the measurement. Cosmological measurements seem to be promising (akin to looking for footprints in the cosmic background from extremely high-energy phenomena from the early universe). And excitingly, extreme precision "table top" experiments are also promising. Check stuff like the recent work on measuring the (nonexistent) dipole moment of the electron.