  We consider approximation of functions of $s$ variables, where $s$ is very
large or infinite, that belong to weighted anchored spaces. We study when such
functions can be approximated by algorithms designed for functions with only
very small number ${\rm dim^{trnc}}(\varepsilon)$ of variables. Here
$\varepsilon$ is the error demand and we refer to ${\rm
dim^{trnc}}(\varepsilon)$ as the $\varepsilon$-truncation dimension. We show
that for sufficiently fast decaying product weights and modest error demand (up
to about $\varepsilon \approx 10^{-5}$) the truncation dimension is
surprisingly very small.
