Genetic algorithm: What is the expected number of strings that are explored?
As I pointed out in the comments, the primary feature of interest for understanding how a genetic algorithm (or evolution, even) explores the fitness landscape is the fitness function. In this case, you specified an extremely simple fitness function with no epistasis. This was a standard assumption for analytical tractability when biology first started out (i.e. most analyses that have Fisher somewhere in the name probably assume no epistasis), but it is seen as less reasonable now. Hence, if your goal was to gain biological intuition, this question will not provide it. Since landscape with epistasis have qualitatively different dynamics.

Finally, you don't specify a selection function, so to give you a heuristic answer I will assume one that has strong selection. 

First note, that at any given time, the number of genotypes you explore is bounded above by the trivial bound of $GN$. To start building a better bound, lets look at Wilf &amp; Evans (2010) that analyzed a no-epistasis model under strong selection is a sexual population (i.e. with recombination). They showed, that it converges to equilibrium (the all $1$s string in your case) in $\Theta(\log l)$ (they actually showed a tighter characterization with a careful analysis of radix-sort, but I will leave that exploration upto you) generations (I think we have to assume that $N \in \omega(l)$). Thus, you will reach the peak very quickly.

On the way to the peak, you will explore $O(N \log l)$ genotypes. In fact, if the selection strength is very high, then we can assume that the population remains close-to monomorphic in terms of the phenotype specified as "number of ones in the genome" although polymorphic in terms of the genotype. This can give us a bound of about $O(l^2 \log l)$.

Once you reach the peak, strong selection will keep you there, and any mutants will go extinct after each generation. This means that from there on out, the population will just stay at the string $1^l$ with transient (single-generation) mutants of Hamming-distance one.

This gives us a pretty ridiculous bound $\min (O(GN), O(l^2 \log l))$. In other words, as long as you look at generations $G \in \Omega(\log l)$ you will only end up sampling an exponentially small fraction of the genome space $O(\frac{l^2 \log l}{2^l})$.