How many elements does it take to normally generate a group?

According, for example, to the following paper by Gonzales-Acuna

http://www.jstor.org/pss/1971036

the smallest number of elements needed to normally generate a group $G$ is called the weight of $G$. This terminology is confirmed in the book

Algebraic invariants of links

by J. Hillman. I also confirm that the "corank" of $G$ usually denotes the largest rank of a free quotient of $G$.

If $G$ is residually $p$-finite or residually [locally indicable amenable], then the weight of $G$ is bounded below by the $b_1^{(2)}(G)+1$, where $b_1^{(2)}(G)$ denotes the first $\ell^2$-Betti number of $G$.

I conjecture that this is the case for all torsionfree groups, but I do not know how to prove this in general.

What you define as $nr(G)$ is indeed commonly referred to as the weight of a group, often written $w(G)$. In general, computing the weight of a finitely generated group, or even any sensible lower bounds on the weight, is very difficult. Wiegold posed the following problem in 1976:

Is it true that every finitely generated perfect group is the normal closure of one element? (i.e., has weight 1).

This can be found as problem 5.52 in the Kourovka notebook: http://arxiv.org/abs/1401.0300

Wiegold's question is `true' in the case of finite groups, and also solvable groups. See M. Chiodo, Finitely annihilated groups, Bull. Austral. Math. Soc. 90, No. 3, 404-417 (2014). In particular, Corollary 5.7 states:

Let $n > 1$ and let $G$ be a finite or solvable group. Then $w(G) = n$ if and only if $w(G^{ab}) = n$, and $w(G)=1$ if and only if $w(G^{ab})= 1$.

-Maurice