I have read that free groups are the "most general" groups given some generators (from Wikipedia):
The construction of a free product is similar in spirit to the construction of a free group (the most general group that can be made from a given set of generators).
I'd like to know: is a free group the most general possible type of group? That is, are there any groups that cannot be described in group presentation (i.e. with a set of generators and a complete set of relations)? If so, what would such a group look like, and how would it relate to a free group? (Can you get that group by somehow restricting a free group? Is it more general than a free group? Are they not comparable?)