At what point does learning JavaScript (possibly after the basic syntax) get messy enough to the point that programmers dislike it?

I'd say it can become messy during basic syntax itself (consider you get to work on someone's code where variable names make no sense, e.g. var t = 70932; no sense in this variable at all").
Let's come back to your question when it becomes messy. Considering that by basic syntax you are pointing to essential client-side interaction code (like calling some function on button click, getting and displaying input value tc.). After basic syntax, let's say your application needs to be connected with server to fetch data, at this moment you would jump the primary coding border and dive into all those ajax calls, validating data and rendering it to your UI for end-user consumption. Here you (or anyone for that matter) can make the code messy by:

1. Not commenting what each method is supposed to do (just like "t" variable example I gave above).

2. Writing function definition inside the loop. I've seen this many devs doing especially during "success" event handler where they run through the retrieved data and writing event handler inside the loop.

3. Adding dynamic nodes to DOM. It is another terrible thing you will see where (again) after getting server data, to display it on UI, developers start adding dynamic nodes to DOM (nothing wrong in it) but most of the time they ignore the fact that where these elements are getting attached. Where in DOM hierarchy they are supposed to be, about their sibling's nodes etc. and this ignorance comes back when your data starts appearing awkwardly on UI and dev start debugging CSS (setting position, z-index, margin etc.)

4. Not respective MVC guidelines. I feel that at times, the code becomes complicated because of situations like project escalations, early release, untimely delivery where the developer is forced to do the work in a hurry. That's where developers start ignoring the fact that not writing correct code at right place might backfire eventually (and most of the time it does happen, speaking by experience). Keep your objects, UI and logic separately. It helps a lot!

5. Naming conventions are ignored (most of the time). As project life increases, team size and faces change. Old people get shifted to other projects, or they leave, and new minds take place. A couple of KT (Knowledge Transfer) sessions are never enough to understand every bit of the code, and the only favour a new coder expects from the old one is to get logical naming conventions, be it a file, variable or function.

6. Last but not least, ignoring the power of "strict mode", these two words alone can do what lines and lines of code can't do; it tells you your sin's beforehand.

There are many more factors which cause messy code and practically not all of them can be thought of, but above are the essential parameters which if prevented can help you live quite easy life at your workplace.

I hope this helps!