There are two different phenomena you may be describing: specialization and network effects.
A nation can only produce a certain amount of things. The common example is the "guns or butter" debate, where a nation must balance the production of guns (military goods) or butter (consumer goods). A nation would be most efficient if it could produce only one, but it requires both. By itself, the nation must choose an inefficient solution - producing both guns and butter, but in smaller quantities than it could if it specialized.
However, if there are multiple nations each one can specialize, one producing guns and one producing butter. They can trade their produce, taking advantage of each others' specialization. The result is that both nations will end up with more of both goods than if they had worked alone. For more information, read about comparative advantage.
I have never heard of a purely political phenomenon which benefits from comparative advantage. There are countless economic examples, much of which is aided (or hindered) by public policy. For example, a state could decide to provide incentives to universities to focus on automotive technology. This leads to specialization, as more automotive experts are trained, more firms founded, and more products created or sold. Other nations no longer need to design or produce cars (although some will want to), and everyone benefits from the trade.
Aside from ordinary trade, some things increase in value as more people use them. This matches your example of the chat app: the more people who use the app, the more value it has for each user.
These are called network externalities (or network effects). These are common in international relations.
Here are a few quick examples. After a few, it should become easy to think of several more:
- International Law: The more nations that agree to be bound by international law, the more value those laws have. For example, one aspect of the Geneva Conventions is the requirement that prisoners of war be treated humanely. If only one or two countries agreed to this, it would have very little value because there would be little assurance that their enemy would be one of the signatories. However, if nearly all of the nations agree to this law, then it becomes significantly more valuable because each nation could reliably be assured that their captured soldiers will be treated well.
- Open Borders and Markets: These kinds of agreements remove borders to people, goods, and capital. The fewer borders there are, the more freely they can move to where they are valuable. So the more signatories to these kinds of deals, the more freely things can move around, and the more value there is.
- Membership in International Organizations: How important would NATO or the UN be if most people choose not to participate? Not very. Membership in international groups increases in value as more people join up.
In all three of these cases, the benefits for each nation increase as the number of participants increases.