Consider reaching a point where you only improve half of what you improved the day before, say first p for some increment p in English proficiency (hence p/2 the next day, p/4 the next…). By definition, you’ll keep improving for the rest of your life, but will never improve 2p counting from that fateful day, à la Zeno. That’s an example, not a model, which I don’t have.

And certainly you have a point, devoting time and money may clearly help vigorously push the upward trend for longer, but I was just talking about my experience, those relatively effortless early moments. In any event, no matter how much you apply, I think it’s hard to argue that it’s quite rare to find foreigners that moved to a country as adults and sound truly native, and I don’t believe it’s due to half-hearted dedication.

It just means one is not improving monotonically. There may be plateaux but the general trend is upwards.