Strongly torsion generated groups are those with a single normal generator, of arbitrary finite order. They are useful for realizing sequences of abelian groups as homology groups. Known examples include stable alternating groups and stable groups generated by elementary matrices. Here the class of such groups is extended, by consideration of other stable classical groups, including orthogonal and symplectic groups. Discussion of other "classical" groups includes a similar result for the stable special automorphism group of a free group. Failure of such a result for mapping class and braid groups is analyzed. It is also shown that the product of finitely many strongly torsion generated groups is strongly torsion generated.