Here are some ruminations which might point in a reasonable direction.
By analogy with how one thinks of power sets, and power sets of powers sets, etc., as higher-order set-theoretic constructions, I would be tempted to term these things "higher-order locales". And in parallel with set theory, a "full" second-order locale would be constructed as an exponential $2^X$ where $X$ is a locale and $2$ is Sierpinski space, if this exponential exists. For if it does exist, then points $1 \to 2^X$ are in bijection with morphisms $X \to 2$, which are in bijection with the original "open sets" of $X$ (that is to say, elements of the associated frame).
It is a theorem that this exponential $2^X$ exists, i.e., that a locale $2^X$ exists such that maps $Y \to 2^X$ are in natural bijection with maps $Y \times X \to 2$ (naturally in $Y$), if and only if the locale $X$ is locally compact. A full account is given in Johnstone's Stone Spaces. It is relevant for this discussion that if this particular exponential $2^X$ exists, then any exponential $Y^X$ exists.
We might just take a moment to explore the sense in which this leads to locales in the category of locales. Following Lawvere, we may define the notion of internal frame as follows: let $E$ be any category with arbitrary small products. Then a frame object in $E$ is a product-preserving functor
$$FreeFrame^{op} \to E$$
where $FreeFrame$ is the Kleisli category for the monad on $Set$ whose algebras are frames, i.e., the full subcategory of $Frame$ whose objects are free frames on sets (the free frame on a set is the frame whose elements are downward-closed sets in the poset of finite subsets of $X$). For an exposition of this general point of view, one could try the nLab article on algebraic theories, which is basically a working out of the theory of infinitary Lawvere theories (which you may well know already, David, since you were a student of Linton and he was really a founding father here).
Anyway, one way of constructing frame objects in $E$ is by constructing product-preserving functors $F: Frame^{op} \to E$, because the composite
$$FreeFrame^{op} \hookrightarrow Frame^{op} \stackrel{F}{\to} E$$
is then also product-preserving. Now of course $Frame^{op} \simeq Loc$; thus we are here contemplating product-preserving functors
$$F: Loc \to E$$
and in particular, we can consider the case where $E = Loc$. Then, if $X$ is a locally compact locale, the exponential functor
$$(-)^X: Loc \to Loc$$
is product-preserving (in fact, it is a right adjoint to $- \times X$, hence preserves all limits). One can use this as a stepping-stone to construct other frame objects in $Loc$ since the category of frame objects in $Loc$ is complete and cocomplete. (And then, of course, we define $Loc(E) = Frame(E)^{op}$, and so we have some techniques for constructing, as a special case, localic locales.)
People do like to play with second-order locales to prove various interesting results. Here is an interesting paper by Martin Escardo (written 'spatially', but I believe Martin likes to think 'localically'), which develops the idea that Theo touched upon in his comment under the question, on compactly many intersections of opens. Johnstone's Elephant also utilizes such techniques, as in the proof of C3.2.8 (which I learned from Mike Shulman under this Math Overflow questionquestion).
I've not thought about transfinite iteration of frame-internalization.
