The only think that comes to mind is "in-house" scripting languages for games. They are generally created only for the game or engine in which they are used. For example; HamsterSpeak (http://rpg.hamsterrepublic.com/ohrrpgce/HamsterSpeak_Specification) was designed for the OHRRPGCE engine and as much as it is the perfect language for that task, you couldn't use it for something else.
Because many games, including big ones like Skyrim, use custom languages, I could say LOP already has a position in the programming world. However, it is a small position compared to OOP and I highly doubt it will grow much bigger then that.
On second thought, HTML could fall under the LOP category. correct me if I am wrong.
Going off of Oria's in-house scripting example, I don't see anyway it would become as big as OOP. The main reason behind my thinking is that it would take a lot of time to design them special in house languages and make them perfect and then they would only be useful for that specific project. So I can only really see them being useful for large projects where in the long run it will pay off. But for small to mid sized and even some large projects it is just a waste of time. Its kind of like reinventing the wheel, for some projects it might pay off to have a more specialized wheel that fits your needs better, but for most you are just wasting time and money doing it.
That is my opinion at least and I am not very familiar with the subject.
@Oria: I'd count HTML as definitely being LOP; it's certainly not general purpose like XML and JSON. It's also more of a serialzied format than a language, IMO, though it looks like they want to disprove the disbelievers with HTML5.
The reason I'm not sure is because; it is a markup language, however it is for a sole purpose and fits under the description of "domain-specific language". So I don't know if that can fall under LOP or if its just something very similar yet different.