The stringy Euler number and stringy E-function are interesting
invariants of log terminal singularities introduced by Batyrev. He
used them to formulate a topological mirror symmetry test for pairs of
certain Calabi-Yau varieties and to show a version of the McKay
correspondence. It is a natural question whether one can extend these
invariants beyond the log terminal case. Assuming the minimal model
program, we introduce very general stringy invariants, associated to
"almost all" singularities, more precisely, to all singularities that
are not strictly log canonical. They specialize to the invariants of
Batyrev when the singularity is log terminal. For example, the
simplest form of our stringy zeta function is, in general, a rational
function in one variable, but it is just a constant (Batyrev's stringy
Euler number) in the log terminal case.