The "smallest" and "Largest" finitely generated infinite group. The smallest finite group that can be generated by $n$ elements and cannot be generated by any less than $n$ elements is a product of $n$ cyclic groups of order $2$.
(a) Is there a largest finitely generated infinite group that can be generated by $n$ elements but not by more than $n$ elements? 
Largest in the sense that if you remove or change any relation between the generators you end up with a group that can be generated by more than $n$ elements. 
ADDED: (a) is not really a question since any infinite group $G$ can be infinitely generated $<G>$.  I'm stupid.
(b) Is there a smallest finitely generated infinite group generated by $n$ elements yet cannot be generated by less than $n$ elements? Smallest in the sense that any extra relation imposed on the group's generators will result in a finite group (or an infinite group generated by less than $n$ elements).
 A: Question (b) is equivalent to asking if there exists a group which is generated by $n$ elements (and no fewer) such that every non-trivial normal subgroup has finite index.
You can always add a group relation which is already implied by the group axioms or existing relations without changing your group. When you impose a non-trivial relation, you are quotienting by the normal closure of all of those elements which satisfy that relation.
There do exist such groups. The easiest example is $\Bbb Z$. All of its nontrivial quotients are finite, and it cannot be generated by 0 elements. Another example is the infinite dihedral group $D_\infty$ which requires 2 generators. In this group all normal subgroups have finite index, but there are still subgroups with infinite index. Also, any finitely-generated infinite simple group satisfies this requirement as mentioned in the comments. If we take a finite generating set, we can reduce it to a minimal generating set. We can then consider the collection of all minimal generating sets and then set $n$ to be the minimum of those cardinalities.
If you only want to consider quotients which have fewer generators than the original group, that seems to be a difficult question. In finitely-generated non-abelian groups, you can have minimal generating sets of different cardinality. And you can even have subgroups which require more generators than the whole group itself does.
