The union of all coreflective Cartesian closed subcategories of $\mathbf{Top}$

The relevant paper is Cartesian closed coreflective subcategories of the category of topological spaces by Juraj Cincura. The first line of the abstract says "Answering the first part of Problem 7 in [10] we prove that there is no largest Cartesian closed coreflective subcategory of the category Top of topological spaces and the same result for the category Haus of Hausdorff spaces."

Another great paper, by Vogt, is Convenient Categories of Topological Spaces for Homotopy Theory. It gives several examples, and argues the pros/cons of each. The author suggests that the coreflective subcategory generated by the locally compact spaces is large enough to contain everything he is interested in.

I contacted Juraj Činčura and he kindly wrote back and directed me to the following observation that is a consequence of results in the paper that David White noted in his answer.

Cartesian closed coreflective subcategories of the category of topological spaces, Topology and its Applications 41 (1991) 205-212.

For an infinite cardinal $a$, let $C(a)$ be the space defined on set $a\cup \{a\}$ where $V\subseteq C(a)$ is open if and only if $V\subseteq a$ or if $a\in V$ and $|a\backslash V|<a$.

Činčura's Proposition 2.1 states that the coreflective hull of $C(a)$ in $\mathbf{Top}$ or $\mathbf{Haus}$ is Cartesian closed. However, Proposition 3.1 states that if $a$ is a strictly larger cardinal than $b$ and class $\mathscr{B}$ contains both $C(a)$ and $C(b)$, then the coreflective hull of $\mathscr{B}$ in $\mathbf{Top}$ or $\mathbf{Haus}$ is not Cartesian closed. Since $C(a)$ and $C(b)$ are both quotients of $C(a)\times C(b)$, this implies that if $a>b$, then any coreflective subcategory in $\mathbf{Top}$ or $\mathbf{Haus}$ containing the space $C(a)\times C(b)$ is not Cartesian closed.

Hence, there is no coreflective Cartesian closed subcategory of $\mathbf{Top}$ or $\mathbf{Haus}$ containing $C(a)\times C(b)$ if $a\neq b$.

This is a specific example showing that there is no largest coreflective Cartesian subcategory of $\mathbf{Top}$ or $\mathbf{Haus}$.