From the <a href="https://ncatlab.org/nlab/show/convenient+category+of+topological+spaces">nLab</a> (although I was the author of these words): 

> A reasonably large class of examples, including the examples of compactly generated spaces and <a href="https://ncatlab.org/nlab/show/sequential+topological+space">sequential spaces</a>, is given in the article by Escard&#243;, Lawson, and Simpson (<a href="https://www.cs.bham.ac.uk/~mhe/papers/ELS03.pdf">ref</a>). 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 (Escard&#243;, 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.