Learn English – She’s always knowing something she’s not supposed to

Sounds perfectly fine to me, though I can see why you are asking. The reason why progressive aspect is not normally used with verbs such as know is that normally they already express a state of continuity on their own and just don't need this. Using them with progressive aspect is normally redundant.

But not in this case. Here we are dealing with a progressive that expresses repetition, not continuity in the strong sense. (See the Wikipedia article. It's currently the last example under "Common uses".) This repetition is in no way inherent in the verb know, and therefore a priori (ignoring the pragmatics discussed in the next paragraph) you should stress it by using the progressive if and only if you would do the same if you used the alternative formulation find out about something.

On the other hand, the way we analyse grammar is not always so logical. Many of us (or just some of us? see discussion) have an internal censor who applies rules such as "know doesn't take progressive aspect" mechanically. This can lead to some native speakers outright rejecting the construction. Even for the others, the rarity of the progressive of know makes the construction stronger. For these two reasons it is best to use the repetition progressives of know and similar verbs more sparingly than those of other verbs - only when you really want to stress the repetitions because they frustrate you. The fact that most native speakers get this right intuitively contributes further to the rarity of the progressive of know. (Some statistics.)

PS: The following example provided by Rathony is even more tricky:

She's always knowing something she's not supposed to know.
^?She's always knowing something she's not supposed to be knowing.

Suppose we have already decided that the first progressive ("always knowing") is just what we want to say in the current context. Now how about the second? Should we use "supposed to know" or "supposed to be knowing"?

At first I wasn't completely sure, but after some analysis I would say "supposed to be knowing" is very likely ungrammatical, and even if it were grammatical, then it would only be appropriate to express such a great deal of frustration that grammaticality probably wouldn't matter anyway.

Now it turns out that the verbs in relative clauses such as "[that] she's not supposed to know" generally don't automatically agree with the main clause verbs (with respect to tense and aspect). This is because the coupling between main clause and relative clause is too loose to permit this. Conversely, non-defining relative clauses are generally used as if they were fully equivalent to the two main clauses that result from a purely mechanical substitution. The relative clause in question is defining, but let's pretend it isn't and make the substitution anyway:

She's always knowing something. She's not supposed to know it.
She's always knowing something. ^?She's not supposed to be knowing it.

Since the infinitives in the second sentences may still be complicating matters, let's get rid of them:

She knows it. But this is not right.
^*She is knowing it. But this is not right.

The first sentence in the second example is ungrammatical. (That does not mean that it is wrong under all circumstances. Most rules of grammar can be broken when we use language creatively in order to express very unusual thoughts such as the state of mind of an Alzheimer's patient. "Right now she is knowing that she never had a child, but yesterday she wasn't. She thought I was her grandson." I wouldn't do this in a single sentence because it's too distracting, but in a longer text about the same person it might be appropriate to do it each and every time.)

But it doesn't follow automatically that "She's always knowing something she's not supposed to be knowing" is ungrammatical. It took some effort to find out that technically it shouldn't be grammatical, and the way language works, the various red herrings taken together make it more likely that someone would say this, and less wrong if they do. Add to this the exasperation of someone who has to witness that she keeps knowing what she shouldn't know, and a single progressive just may not feel strong enough. It is then natural to make the relative clause agree in aspect so that it also carries the frustration. (Even though technically it has nothing to do with it.)

If people talk like that often enough, it becomes grammatical. The progressive of know is probably too rare to test this, but we could try to test it with sentences like this:

She is always eating things she shouldn't eat.
^?She is always eating things she shouldn't be eating.

We could search for similar sentences in a large corpus and for each try to measure the degree of frustration and find out the scope of the statement in the relative clause. Examples:

She is always eating things she shouldn't eat. Socks, paper clips, once even a screwdriver.
She is always eating things she shouldn't be eating. But eating my passport at the airport gate? This time she's gone too far!
Whether she is pregnant, on a diet or preparing the buffet for the customers: She is always eating things she shouldn't be eating!!

If the result of this hypothetical research project is that a progressive in the relative clause is often caused by just the frustration that also causes it in the main clause, then we should probably consider "She's always knowing something she's not supposed to be knowing" grammatical. If it's too rare, then it's not grammatical. But since ungrammatical sentences can actually be another way of expressing an agitated state of mind, it turns out that under the very circumstances when it might be appropriate, grammaticality doesn't really matter anyway. (This is of course not an accident.)