What is so amazing about having least upper bound (and dually, the greatest lower bound)? Why are these properties so important, that they spawned lattice theory? Or, why were posets having these properties (lattices) identified as being of special interest compared to other posets?
 A: I started studying lattices, because I wanted to better understand certain hierarchical structures that arise when studying the dependency structure between equations, variables and derivatives in differential algebraic systems by combinatorial means. A simple approach is to represent the dependency structure as a bipartite graph (where the one set of nodes is given by the equations, the other by the variables, and an edge is there when the variable occurs in the equation), compute a maximum matching in that graph, and derive a partially ordered block structure from that matching like the Dulmage–Mendelsohn decomposition (in case of a perfect matching, this is just the strong component decomposition of the directed graph, which arises from identifying the matched nodes). Well, that simple approach actually only treats the dependency structure of a system of equations, but similar (although more complicated) techniques can handle the differential algebraic dependency structures. This and much more is explained in books like Matrices and Matroids for System Analysis.
The book title contains the magic word "matroid". I first tried to better understand matroids, but these are strange beasts. They somehow lead to non-trivial efficient algorithms, but still manage to dodge an intuitive understanding. But matroids are related to lattices and orders, see for example the definition of Matroid in terms of closure operators. 
Two typical things of my case seem to be that the initial questions arose in a finite combinatorial context, and for structures "less general" than lattices.

Three main themes for lattice theory might be abstracting essential features from "systems of subsets", the "category theory" of categories where each hom-set has at most one element (preorder), and algebraic models corresponding to certain logics. There are more themes than this, for example the intimate connection between universal algebra and lattice theory. Or the "structure derived from maximum matching in a bipartite graph" questions, as investigated by formal concept analysis.
For the "system of subsets" theme, there are two different hierarchical structures. First there is the subset ordering. If the system is closed under finite unions and intersections, the resulting lattice will be distributive. Some systems like the open subsets of a topology are not closed under arbitrary intersections, and the corresponding lattice theoretic concepts leads to Pointless topology. But dually, there are also the partitions of a given set $U$, which can be represented as equivalence relations. These are subsets of $U\times U$, and they are closed under intersection. They are not closed under unions, but the least upper bound is still unique and always defined. The resulting lattice is in general not distributive. But for typical (=commuting) equivalence relations derived from congruences of algebraic structures, it is still modular (and also arguesian).
For the "category" theme, the conditions for a cartesian closed category basically lead to a Heyting algebra structure. But as you only asked about "least upper bound", this is equivalent to the existence of all binary coproducts in the corresponding category. A residuated lattice probably arises from the conditions for a closed monoidal category, but I haven't checked.
For the "logic" theme, you can study Boolean algebras as lattices. Apart from Boolean algebras and Heyting algebras, I'm not sure how far the "logic" themes has really been explored. In a certain sense, the connection to universal algebra is also a connection to first order logic, but this seems to be somehow the wrong direction.

As a conclusion, let me say that although there are many things one can study in relation to lattice theory, I'm not sure whether lattice theory is really the "correct" unifying framework. For all these things, there is a partial order with all finite least upper and greatest lower bounds, but so what? It doesn't feel like lattice theory would be a unifying framework abstracting the essential features related to order and hierarchy. Contrast this with category theory, where I and many other people are convinced that it is a unifying framework abstracting the essential features related to functions and correspondences. Edit: Because user89 stressed that this paragraph makes sense to him, I asked myself why lattice theory doesn't feel like it would abstract the essential features. There is some sort of asymmetry in most instances of order and hierarchy, like the open and closed sets in a topological space, the difference between $f(x)\neq y$ and $f(x)=y$, or the nice properties related to universal Horn-clauses. While category theory with its directed arrow turns these asymmetries into nice dualities, lattice theory somehow seems to deny the very existence of these asymmetries. But often, I really would like to be able to nail down these asymmetries. For example, even the equations and variables from my introductory example are not as interchangeable as it seems. An equation often corresponds to something fundamental, like energy conservation, but a variable is often just an artifact of an arbitrarily chosen coordinate system.
