  We consider particle collisions in the background of a nonextremal black
hole. Two particles fall from infinity, particle 1 is fine-tuned (critical),
collision occurs in its turning point. The first example is the
Reissner-Nordstr\"{o}m (RN) one. If the energy at infinity $E_{1}$ is big
enough, the turning point is close to the horizon. Then, we derive a simple
formula according to which $E_{c.m.}\sim E_{1}\kappa ^{-1/2}$, where $\kappa $
is a surface gravity. Thus significant growth of $E_{c.m.}$ is possible if (i)
particle 1 is ultrarelativistic (if both particles are ultrarelativistic, this
gives no gain as compared to collisions in flat space-time), (ii) a black hole
is near-extremal (small $\kappa $). In the scenario of multiple collisions the
energy $E_{c.m.}$ is finite in each individual collision. However, it can grow
in subsequent collisions, provided new near-critical particles are heavy
enough. For neutral rotating black holes, in case (i) a turning point remains
far from the horizon but large $E_{c.m.}$ is still possible. Case (ii) is
similar to that for collisions in the RN metric. We develop a general
theoretical scheme, direct astrophysical applications can be a next step to be
studied.
