From the nLab (although I was the author of these words):
A reasonably large class of examples, including the examples of compactly generated spaces and sequential spaces, is given in the article by Escardó, Lawson, and Simpson (ref). These may be outlined as follows. An exponentiable space in $Top$ is a space $X$ such that $X \times -: Top \to Top$ has a right adjoint. These may be described concretely as core-compact spaces (spaces whose topology is a [[continuous lattice]]). Suppose given a collection $\mathcal{C}$ of core-compact spaces, with the property that the product of any two spaces in $\mathcal{C}$ is a colimit in $Top$ of spaces in $\mathcal{C}$. Such a collection $\mathcal{C}$ is called productive. Spaces which are $Top$-colimits of spaces in $\mathcal{C}$ are called $\mathcal{C}$-generated.
Theorem (Escardó, Lawson, Simpson): If $\mathcal{C}$ is a productive class, then the full subcategory of $Top$ whose objects are $\mathcal{C}$-generated is a coreflective subcategory of $Top$ (hence complete and cocomplete) that is cartesian closed.
Other convenience conditions, such as inclusion of CW-complexes and closure under closed subspaces, are in practice usually satisfied as well. For example, if closed subspaces of objects of $\mathcal{C}$ are $\mathcal{C}$-generated, then closed subspaces of $\mathcal{C}$-generated spaces are also $\mathcal{C}$-generated. If the unit interval $I$ is $\mathcal{C}$-generated, then so are all CW-complexes.
A number of examples are scattered throughout the paper.
