  We study acts and modules of maximal growth over finitely generated free
monoids and free associative algebras as well as free groups and free group
algebras. The maximality of the growth implies some other specific properties
of these acts and modules that makes them close to the free ones; at the same
time, we show that being a strong "infiniteness" condition, the maximality of
the growth can still be combined with various finiteness conditions, which
would normally make finitely generated acts finite and finitely generated
modules finite-dimensional.
