A category with weak equivalences which is not a model category I'm only considering complete and cocomplete categories. A pair $(\mathfrak{X} , \mathfrak{W}) $ is, by definition, a category with weak equivalences if  $ \mathfrak{X} $ is a category and $ \mathfrak{W} $ is a subcategory satisfying the $2$ out of $3$ axiom.
I was wondering if there are nice examples of categories with weak equivalences for which there is no model structure. 
Thank you
 A: Here is another (more combinatorial) answer.  Let $\kappa$ be any (nice) cardinal, and let $\mathcal{A}$ be the union of the full subcategory of $\mathbf{Set}$ consisting of all sets of cardinality less than $\kappa$ and the subcategory of all isomorphisms in $\mathbf{Set}$.  $\mathcal{A}$ is closed under retracts and 2-of-3, but it is NOT the subcategory of weak equivalences of a model structure on $\mathbf{Set}$.  We can see this just by listing all of the (nine) model structures on $\mathbf{Set}$, and noting that the subcategories of weak equivalences are


*

*all morphisms

*all isomorphisms

*all morphisms between nonempty sets (plus the identity on $\emptyset$)


(See Tom Goodwillie's answer to https://mathoverflow.net/questions/29653.) In particular, model categories can determine whether you're empty or nonempty, but they can't differentiate between different set sizes.
A: A very interesting example: consider semi-simplicial sets (alias $\Delta$-sets).
These are simplicial sets without degeneracies, and there is an ``adjoin degeneracies'' functor from semi-simplicial sets to simplicial sets that is left adjoint to the evident forget degeneracies functor.  One can compose this with geometric realization or one can define geometric realization of semi-simplicial sets directly.  Define a weak equivalence of semi-simplicial sets to be a map whose geometric realization is a homotopy equivalence (weak equivalence is the same since these are CW complexes).  Since it is a presheaf category, the category of semi-simplicial sets is bicomplete, and I've described the obvious weak equivalences, which satisfy the two-out-of-three property.  Matthew Thibault convinced me that this truly natural category with weak equivalences does not admit a model structure for any choice of cofibrations or fibrations.
EDIT: Dylan, if you believe the comments already posted, you know the proof. 
According to Tom, the acyclic fibrations have to be the isomorphisms.  But 
then all maps are cofibrations, since they are the maps with the LLP wrt the
acyclic fibrations.  If you believe Karol that it cannot be the case that all
monomorphisms are cofibrations, or just that $\ast$ cannot be cofibrant, you already 
have a contradiction to such a model structure.
