A pair $(\mathcal{A}, \mu)$ of a $\sigma$-complete boolean algebra $\mathcal{A}$ and a functional $\mu : \mathcal{A} \to [0, \infty]$ is called a measure algebra if $\mu$ is strictly positive and countably additive on disjoint sequences. A measure algebra is semifinite if whenever $\mu(a) = \infty$ there exists a $b < a$ such that $0 < \mu(b) < \infty$. A measure algebra is localizable if it is complete and semifinite.
A measure algebra can be constructed from a measure space by taking the boolean algebra of equivalence classes of measurable sets modulo the null sets. In the reverse direction, by the Loomis-Sikorski Theorem, every $\sigma$-complete boolean algebra is isomorphic to the quotient of the $\sigma$-algebra $\{ A \bigtriangleup B : A \text{ clopen }, B \text{ meager } \}$ by the $\sigma$-ideal of meager sets. An ordinary measure can be defined on the quotient in the natural way, and the concrete measure algebra of the resulting measure space is isomorphic to the original measure algebra.
The preceding construction can also be used to show that the localizable measure algebras are precisely the Boolean algebras of projections of commutative von Neumann algebras. Note the similarities between the definitions of a localizable measure algebra and a normal semifinite weight on a von Neumann algebra.
Due to the semifiniteness of the measure, the problem of characterizing the Boolean algebras that are measure algebras can be reduced to the case of finite measure. Call a $\sigma$-complete boolean algebra $\mathcal{A}$ finitely measurable when there exists a functional $\mu : \mathcal{A} \to [0, \infty)$ making it a measure algebra. Then a complete boolean algebra $\mathcal{A}$ has a functional $\mu : \mathcal{A} \to [0, \infty]$ making it a localizable measure algebra precisely when the set $\{ a \in \mathcal{A} : \mathcal{A}_a \text{ is finitely measurable } \}$ is order-dense in $\mathcal{A}$, where $\mathcal{A}_a$ is the principal ideal generated by $a$.
Unfortunately, even in the finite measure case there is no great solution to this problem. There’s a characterization by Kelley that reduces it to the existence of a strictly positive finitely additive measure and a combinatorial condition (what he calls weakly countably distributive). He also characterizes the existence of a finitely additive measure in terms of intersection/covering numbers. Gaifman wrote a survey paper on this problem, and Jech proved a game-theoretic characterization.
A good reference for most of the facts mentioned about measure algebras is volume 3 of Fremlin’s tome Measure Theory, particularly chapters 32 and 39.
First, if you haven't already you should have a look at this introductory paper by P.T. Johnstone The Art of pointless thinking which gives a lot of insight on how locale theory works.
Here are some observations which I hope will answer your questions:
1) As I said in the comment, when you glue locales together along open subspaces in a way similar to affine schemes, the objects you get aren't new, they are just bigger locales so there is no need for new objects (the missing corner is "locales" again).
2) In this picture, toposes are more like Stack/groupoid objects. The analogy is not perfect; more precisely toposes are "kinds of stacks" but do not correspond to those one consider in algebraic geometry like the Artin stacks or the Deligne Munford stacks. This point of view on toposes is I think most visible in Marta Bunge An application of descent to a classification theorem for toposes (I haven't found a freely available version) but also appears in some Paper by I. Moerdijk and of course all of this is a consequence of the amazing and famous Joyal & Tierney "An Extension of the Galois Theory of Grothendieck" (which is unfortunately not that simple to find). In this sense locales are the building blocks for toposes.
3) Locales are indeed a special case of toposes; they corresponds to the "localic toposes" which are exactly the toposes that are generated by subobjects of the terminal object. This notion of localic topos can be promoted to a notion of localic morphism and is rather important in topos theory (basic theory of this notion can be found in A.4.6. of P.T. Johnstone's Sketches of an Elephant).
(This paragraph is not meant to be formal.) To some extent the idea of topos is an extension of the idea of locale: in topos theory you are trying to do topology not with open subsets as the basic objects, but with sheaves as the basic objects. A sheaf (of sets) on a space is always obtained by gluing open subspaces together (along open subspaces) so in some sense sheaves are a generalization of open subspaces. From this perspective locales are just the toposes whose "topology" can be generated by open subspaces (the use of the word topology here is informal and has not much to do with Grothendieck topology).
4) To come back to the question of seeing locales a some sort of geometric objects. As I already said in the comment, such a general point of view of people working with locales is that locales ARE geometric objects by themselves, somehow more fundamental than topological spaces. It's not locales that are "structured topological spaces" -- it's topological spaces that are structured locales (a topological space is the data of a set of points $X$, a locale $L$, and a surjective map of locales from $X$ to $L$, so they are just locales with a specific set of points marked).
5) From the discussion in the comments I get that what troubles you is to consider as geometric an object which might have not any points; I will try to address that in the end of my answer.
First, as you probably know, there is an equivalence of categories between locales having enough points (it mean enough points to distinguish the elements of the frame) and sober topological spaces. So in a first approximation one can consider that locales without (enough) points and non-sober topological spaces are pathological objects, and that except for those pathologies, locales and topological spaces are the exact same things. That is what people were doing for quite some time (the first example of point-free locales/toposes were considered as being completely pathological objects).
It appears that there is a lot of very interesting example of locales without points and that those are not that pathological after all, but this has been realized more recently. A good example of that is that there is a sublocale of $\mathbb{R}$ which is the natural domain of definition of functions that are defined "almost everywhere": as such functions cannot be evaluated at any specific point, this locale cannot have any point. I think this example is studied in length in Alex Simpson Measures, Randomness and sublocales
6) The main interest of locale theory is to realise that topology doesn't really care about points, and to some extent works better if one does not care about points. So trying to construct faithful functor from the category of locales to the category of sets is a little bit weird from this perspective. I could have answered to your comment " It is intuitive that continuous maps map points to points" that it is also the case with locales that points are sent to points, but it's just that we don't really care about points.
In fact, before Cantorian set theory which pushed everyone in mathematics to think about everything as a set, most mathematicians were not thinking about spaces like the real numbers as being sets + a topology (in the Cantor style view of sets as a discrete objects) but really as a "continuum" not formed of "discrete points". This is the point of view that locale theory is pushing forward.
But if you really want to have a faithful functor from locales to sets, which has some geometrical meaning, here is a way to get one:
If you start from a frame $A$ you can see it as a distributive lattice and attached to it its Stone spectrum Stone(A) which is the compact (in general not Hausdorff) space of prime filters of $A$. Morphisms of frames are a special case of morphisms of distributive lattices, so this produces your faithful functor and I think it is the most geometric one can come up with.
The points of the corresponding locales $Loc(A)$ are the totally prime filters of $A$: "prime filter" means if $a \cup b \in P$ then $a \in P$ or $b \in P$, while "totally prime filter" means if $ \bigcup a_i \in P$ then $\exists i, a_i \in P$. So the points of the locale form in some sense a specific subspace of $Stone(A)$.
From the localic point of view this can be promoted to a morphism of locales $Loc(A) \rightarrow Stone(A)$, and it is always the case that the locale $Loc(A)$ is a dense subspace of $Stone(A)$, but of course $Loc(A)$ can have no points. You can think of points of $Stone(A)$ as an approximation to (eventually non existing) points of $Loc(A)$. But I am not sure this picture gives the correct insight on locales: for example if you applied this to the locale corresponding to an ordinary topological space (like the real number) you will get a very big and not very natural space $Stone(A)$ which is a lot more complicated than the space you started with.
Best Answer
First off, let's forget the transfinite iteration. As David said, one often interprets algebraic objects such as "group objects" in sufficiently nice categories, intending the word "group" as a cipher for any kind of algebra, so let's also say "space" for any object of any kind of "sufficiently nice" category. Monads provide a notion of "algebraic theory" that can be interpreted in any category.
So we can formulate the question more abstractly by asking for a category $S$ (whose objects we call "spaces") for which the opposite category $S^{op}$ is equivalent to the category of algebras for a monad on $S$ (and we call these algebras "frames").
This is exactly the idea behind my Abstract Stone Duality research programme, which was so called because both the duality between algebra and geometry and the "always topologize" slogan behind David's question were due to Marshall Stone.
In the ASD programme I developed this idea in the case where the adjunction between $S$ and $S^{op}$ is given by the exponential $\Sigma^{-}\dashv\Sigma^{-}$, where I write $\Sigma$ rather than Todd's 2 for the Sierpinski space. These ideas are summarised from a foundational point of view in Foundations for Computable Topology. Mathematically, the pay-off of the monadicity hypothesis was a theory of computable analysis that satisfies the Heine-Borel theorem, for which see The Dedekind Reals in Abstract Stone Duality with Andrej Bauer. This was applied to the intermediate value theorem in A Lambda Calculus for Real Analysis, which paper is the best introduction to ASD for ordinary mathematicians.
This idea works properly for Computably Based Locally Compact Spaces.
Can we do something similar taking $S$ to be the category of locales, to get back to David's question? Indeed, Steve Vickers has studied this, using his double powerlocale monad. (I find this easiest to understand as the comonad on frames that arises from the forgetful functor to dcpos.) He uses the name localic locale an object of the opposite of the category of algebras for the monad over locales (or a coalgebra for the comonad on frames), although he has also called them colocales and I like this name myself.
We therefore have the categories $L$ of locales and $C$ of colocales, where $C^{op}$ is monadic over $L$, but they are not equivalent. David may therefore ask what the iteration of this construction yields, but in fact it stops at stage 2: $L^{op}$ is also monadic over $C$.
Even so, this cannot be the end of the story, because we would like $L$ and $C$ to be subcategories of a single cartesian closed category with finite limits. Steve has used the presheaf category for this and shown that the monad is actually the double exponential in this sense. Reinhold Heckmann constructed a smaller category of equilocales. However, the structure of these two categories is far more complicated than that of equilogical spaces.
I am currently studying another categorical idea called equideductive logic, the slogan for which is "a category that lies nicely within its cartesian closed extensions". As in Steve's work, the double exponential brings you back into the smaller category, but I also ask that any subspace of an object in the smaller category (ie an equaliser targeted at any object of the larger category) also lie in the subcategory.