I would like to know about your favorite group(s). Since groups do appear everywhere in mathematics and there are plenty of them, which ones have drawn your attention the most or surprised you? Please not just name the group, but also provide some facts about it why you find this one particularly interesting.
I'll start by mentioning Grigorchuks group. Because is was the first group i encountered, wich is finitely generated but not finitely presented. Also it was the first group discovered with intermediate growth. It has a lot of "strange" properties like:
it's infinite but residually finite
it's amenable but not elementary amenable
every proper quotient group is finite
every maximal subgroup has finite index
Also Grigorchuks group acts as a key-counterexample in infinite group theory. My professor once told me: "If you have a conjecture about infinite groups, try it one Grigorchuks group. If it holds, it might be worth trying to prove it."
edit: flagged for community wiki.
What about the smallest non-trivial group $\mathbb{Z}/2\mathbb{Z}$ ? See the discussion Fantastic properties of $\mathbb{Z}/2\mathbb{Z}$ for many convincing arguments.