(The discussion below is for pointed spaces.) I'll use $\mathcal{F}_*$ for the pointed version of your $\mathcal{F}$.
As Nicholas Kuhn says, this is related to the closed classes studied by E. Dror Farjoun and W. Chach'olski. Including closure under limits in your collection actually makes it closer to the dual concept, which I have studied under the name of "resolving classes" and "resolving kernels".
I can't give a definitive answer to the question as posed, but I can show that the pointed version $\mathcal{F}_*$ contains no Eilenberg-MacLane spaces $K(G,n)$ for $n> 1$ or for $G$ finite abelian and $n = 1$
A resolving kernel is a collection of spaces of the form $$ \mathcal{R} = \{ Y \mid \mbox{$\mathrm{map}_*(X,Y) \sim *$ for all $X\in \mathcal{X}$}\} $$ for some collection of spaces $\mathcal{X}$.
Theorem: Let $\mathcal{A}$ be a collection of pointed spaces such that
- $\Sigma\mathcal{A}\subseteq \mathcal{R}$
- $\mathcal{A}\wedge \mathcal{A} \subseteq \mathcal{R}$
(up to weak equivalence). If $\mathcal{R}$ is a resolving kernel and $\Sigma \mathcal{A} \subseteq \mathcal{R}$, then the closure of $\mathcal{A}$ under "finite-type wedge" and extensions by cofibrations (and in particular, under pushouts) is contained in $\mathcal{R}$ as well.
(A finite type wedge is a wedge in which the connectivities of the summands increases to infinity.)
It is worth pointing (ha ha) out that the pointed case is quite different from the unpointed case, because getting even a single wedge into a resolving kernel is enough to bootstrap to the hypotheses of this theorem. Explicitly:
Theorem: If $X$ is a finite-type space with $\mathrm{map}_*(X, S^n\vee S^m) \sim *$ for any two $n, m > 1$, then $\mathrm{map}_*(X,K)\sim *$ for all simply-connected finite-dimensional CW complexes $K$. (If $\pi_1(X)$ is not a perfect group, then "simply-connected" can be dropped from the conclusion).
Now let's think about $\mathcal{F}_*$, the smallest class of pointed spaces containing all the spheres and closed under
- homotopy pushouts
- homotopy pullbacks
- homotopy retracts.
This is contained in $\mathcal{S}$, the smallest resolving kernel containing all the spheres. Let $\mathcal{M}_p$ denote the resolving kernel associated to $\mathcal{X} = \{ B\mathbb{Z}/p \}$ for your favorite prime $p$. Miller's theorem (the Sullivan conjecture) tells us that $$ \mathcal{F}_*\subseteq \mathcal{S} \subseteq \mathcal{M}_p. $$ But obviously $B\mathbb{Z}/p \not \in \mathcal{M}_p$, so we can conclude $$ B\mathbb{Z}/p \not\in \mathcal{F}_* $$ Similarly, $\mathcal{F}_*$ cannot contain Eilenberg-MacLane spaces $K(G,n)$ for any finite abelian group $G$ (such spaces also cannot see spheres by an easy extension of Miller's theorem). And using closure under forming homotopy fibers, we can see that $\mathcal{F}_*$ cannot contain $K(\mathbb{Z},n)$ for any $n$. Thus we have
Proposition: If $G$ is an abelian group, then $K(G,n) \not\in \mathcal{F}_*$ for $n > 1$.