  If two particles collide near the black hole horizon, the energy in the
centre of mass (CM) frame can grow indefinitely (the so-called the BSW effect).
This requires fine-tuning the parameters (the energy, angular momentum or
electric charge) of one particle. We show that the CM energy can be unbound
also for collisions in the space-time of quasiblack holes - QBHs (the objects
on the threshold of forming the horizon which do not collapse). It does not
require special fine-tuning of parameters and occurs when any particle inside a
QBH having a finite energy collides with the particle that entered a QBH from
the outside region.
