First of all, the scheme of the CERN accelerator complex you
posted contains not only that single chain which brings the
protons to the LHC, but also several other chains which are used
for many lower energy experiments conducted in parallel at CERN.
But let's focus on the LHC accelerator chain: why do we need
several successive accelerators instead of a single one? The
answer is very intuitive from the engineering point of view:
technologically, it is much easier to construct several devices
specialized for different but limited ranges of some physical
parameter than to build a single device that exhibits excellent
performance in the entire vast range. An example in
transportation: cars are good to travel at lower speed,
airplanes are good for traveling fast. It is exceedingly
difficult to build a vehicle that would have equally good
performance in the entire range of speeds from 1 km/h to 1000
km/h. Another example is given by professional acoustic systems
which usually include several speakers optimized fo