For me, a relative path in a require statement signifies that the required code is a part of the application itself.

Consider:

    require('foo');

Versus:

    require('./foo');

If I find that a module is becoming popular, I may turn it into an npm module by moving it to it's own folder and adding a package.json, then using npm's bundledDependencies property to inform npm that it lives locally.

(edit: Note that I've shifted my perspective on the module when this happens. The shift is from "This code is in a separate module to keep my application organized" to "This code is in a separate module because it has some logical independence from the application.")

If the module needed even more independence, I'd move it to it's own git repo and create an internal mirror of the npm repo so that it can be discovered internally, and unbundle it from my app. (I could see this happening especially with larger teams, or teams that embrace the "module all the things" philosophy.)

And of course, there's always the possibility of moving the module out to the public npm repos and hosting it there.

I've never had require with relative paths of more than two or three "double dots". (Note to self: Look up what those things are called formally.)