Is there a category of topological-like spaces that forms a topos? The category of convergence spaces generalise topological spaces and form a quasi-topos, as topoi are allegedly nicer is there a nicer kind of topological-like space, the category of which forms a topos?
 A: Let me point out a couple of other possibilities, they may come handy in certain situations. The realizaibility topos $RT(P(\omega))$ over Scott's graph model $P(\omega)$ contains countably based spaces as a full subcategory. The inclusion preserves countable limits, countable coproducts, and those exponentials which happen to exist in countably based spaces. This can be generalized to spaces of arbitrary weight by using a larger graph model $P(\kappa)$. But as Todd points out, such a topos contains a lot of non-topological junk.
If you do not insist on a topos, and are willing to consider slightly less than a topos, then the exact completion of topological spaces is a good candidate. The objects are topological spaces equipped with a (formal) equivalence relation, so they feel like spaces still. Not too much junk there. For this sort of thing, see

Carboni, A., Rosolini, G. Locally cartesian closed exact completions, J.Pure Appl. Alg., 154, 2000.

A naive way to get spaces inside a topos is to simply consider (pre)sheaves on all spaces, which of course runs against size restrictions. Nevertheless, a useful picture emerges, see

Rosolini, G., Equilogical spaces and filter spaces, Rend. Circ. Mat. Palermo, 64, 2000

A: MR0531162 
Johnstone, P. T.
On a topological topos.
Proc. London Math. Soc. (3) 38 (1979), no. 2, 237–271.
"The author defines a topos E which has the category of sequential spaces F as a reflective subcategory whose inclusion functor preserves the colimit diagram arising from any open cover, certain colimit diagrams arising from closed covers, and the equalizer diagram giving the quotient space for an equivalence relation which is sequentially closed. The author answers the questions which spring to mind for the topos theorist about E:...."
@Todd: The notion of topological space has been around for a long time and we need to consider whether or not it is the "right" concept for various needs. Grothendieck in Section 5 of "Esquisse d'un programme" argues for more elaborate concepts  such as stratifications to model geometry. Another question is whether the category of the objects under consideration has "convenient" properties; in some cases this might be the prior consideration. In any case, the use of this "topological topos" in say algebraic topology should be considered. For example, it might be appropriate in the theory of parametrized spectra, and also develop further Peter Booth's notion and applications of fibred exponential laws. 
A: Ronnie has already given the answer which immediately popped into my head when I saw the question. But I should sound a warning that the objects of the topological topos aren't exactly topological spaces, and indeed it seems likely that the only full subcategories of $Top$ that are toposes are fairly uninteresting for topology: things like the full subcategory of discrete spaces or the full subcategory of codiscrete spaces, or the degenerate topos which consists of only a terminal object. 
We think of topological spaces as sets with extra structure, meaning that the underlying-set functor $\hom(1, -): Top \to Set$ is faithful, i.e., morphisms of $Top$ are functions satisfying some property. Suppose $T$ is a full subcategory of $Top$ whose terminal object is a 1-element space, denoted as above by $1$. Then $\hom(1, -): T \to Set$ is again faithful. This means that if $T$ were a topos, it would be a well-pointed topos. If moreover $T$ were a complete well-pointed (nondegenerate) topos, then it would be equivalent to $Set$. 
We see here a certain tension here between nice categories with some "un-nice" objects (e.g., the topological topos, some of whose objects are not "nice" topological spaces), and categories of "nice objects" which fail to be maximally nice as categories (e.g., toposes). But sometimes one can split the difference; the convergence spaces of the OP seem about as good a compromise as any. 
