This is a fun one! Here are some of my favorite examples (I'll try not to list ones that have already been listed).

Finite Group : SL_n(Z/pZ).

Nonabelian group (simple) : Free groups, Baumslag-Solitar groups

Nonabelian group (hard) : SL_n(Z), Aut(F_n), mapping class group

Lie algebra : Depending on context, either sl_n(C) or a free Lie algebra

Algebraic Variety : twisted cubic, a hyperelliptic curve