It is a lofty goal. I've been told by coworkers, "No one can remember the entire language." and then went and did it anyways.

Yes, I still have to look up libraries for, eg pip, and I haven't gotten to the level of knowing every compiler error yet.

But it is very much a doable goal. Especially given that knowledge transfers between languages and libraries. Once you learn the thought process of the dev who wrote it, you can guess how they did or are going to do everything after that, creating a short cut where you can guess and usually always guess the right answer. It's much quicker than using SO.

> The amount of languages, libraries and APIs we have to regularly deal with

Is this because this Web programming innately needs lots of tools, etc. to work, or because the modern programming ecosystem (including stack overflow) has allowed it to grow to be this way?

I would say a little bit of both. Since modern networked programming is much more complex than before (having to care about security, distributed storage, systems, worldwide connectivity over vast range of devices etc.) it has more moving parts and thus more combinations of various systems possible. Naturally complexity grows exponentially.

>...complexity grows exponentially

Just to nitpick a bit, I would say that complexity grows quadratically - proportionally to the number of potential interactions between subcomponents.

There can be interactions among more than 2 components as well that cannot be explained just by the pairwise interactions. The number of all subsets grows exponentially.