The space parameters are given by \fontdimen2 through \fontdimen4. For the default cmr10 these correspond to a space of 3.33pt plus 1.67pt minus 1.11pt. Computer modern has a fairly large interword space, most others will be smaller. Eg, MinionPro: 2.27pt plus 1.135pt minus 0.757pt.
An overfull hbox will result when the badness of a line is worse than \tolerance, which by default is 200. Badness is defined as 100 times the cube of the glue shrink ratio, so the smallest interword space which would not cause an overfull hbox, with the default settings for all the parameters, is 1.933pt for cmr10, and 1.316pt for MinionPro.
\documentclass[10pt]{article}
%\usepackage{MinionPro}
\begin{document}
Interword space: \the\fontdimen2\font\
plus \the\fontdimen3\font\ minus \the\fontdimen4\font
tolerance: \the\tolerance
Minimum space (assuming tolerance of 200): \the\dimexpr\fontdimen2\font-
1.25992\fontdimen4\font\relax % cube root of (200/100) is 1.25992
\end{document}