  We study the question of finite generation of saturated multi-Rees algebras
and investigate the asymptotic behaviour of related length functions. In the
setup of excellent local domains, we show that the saturated multi-Rees algebra
of a finite collection of ideals is finitely generated when the analytic spread
is not maximal and the associated length function eventually agrees with a
polynomial. Similar results are obtained when we restrict to two-dimensional
local UFDs with no restrictions on the analytic spread. We further prove that
the saturated multi-Rees algebra of finitely many monomial ideals in a
polynomial ring modulo an irreducible monomial ideal, is always finitely
generated. In this case, the corresponding length function is shown to exhibit
piecewise quasi-polynomial behaviour. We also produce multi-ideal versions of a
theorem of Amao.
