  Iterated Bessel processes R^\gamma(t), t>0, \gamma>0 and their counterparts
on hyperbolic spaces, i.e. hyperbolic Brownian motions B^{hp}(t), t>0 are
examined and their probability laws derived. The higher-order partial
differential equations governing the distributions of
I_R(t)=_1R^\gamma(_2R^\gamma(t)), t>0 and J_R(t)
=_1R^\gamma(|_2R^\gamma(t)|^2), t>0 are obtained and discussed. Processes of
the form R^\gamma(T_t), t>0, B^{hp}(T_t), t>0 where T_t=\inf{s: B(s)=t} are
examined and numerous probability laws derived, including the Student law, the
arcsin laws (also their asymmetric versions), the Lamperti distribution of the
ratio of independent positively skewed stable random variables and others. For
the process R^{\gamma}(T^\mu_t), t>0 (where T^\mu_t = \inf{s: B^\mu(s)=t} and
B^\mu is a Brownian motion with drift \mu) the explicit probability law and the
governing equation are obtained. For the hyperbolic Brownian motions on the
Poincar\'e half-spaces H^+_2, H^+_3 we study B^{hp}(T_t), t>0 and the
corresponding governing equation. Iterated processes are useful in modelling
motions of particles on fractures idealized as Bessel processes (in Euclidean
spaces) or as hyperbolic Brownian motions (in non-Euclidean spaces).
