which is the subject of "was possible"?

Monarch butterflies sometimes fly through clouds when they migrate.
It happens; so, it's possible.

That is possible.
She never dreamed [that] that was possible.
That was definitely something [that] she never dreamed was possible.

This "was possible" is a clause missing its subject.  That's the gap which user178049's answer mentions.  In the original, this incomplete clause is inside a contact clause.  I've shown where an optional relative "that" could be used in the sentences above.

The relative pronoun "that" is usually optional.  The most common and most obvious exception is when it is the subject of the subordinate clause which it introduces.  In the last example above, the relative "that" is a pronoun and it is the subject of "was possible", but it introduces "she never dreamed [that] was possible".  The subject of that clause is "she".

We can infer that a relative pronoun is implied because such a pronoun can fill the gap.  We can explain the missing relative pronoun by showing that it is not the subject of the relative clause, although it is the subject of the content clause within the relative clause.

 

Here, you’ll learn how data classes enable you to write code that’s cleaner and more concise than you ever dreamed was possible.

Clean and concise code exists.  As long as there is room for improvement, cleaner and more concise code could exist.  That is to say, it's possible.

Code that is cleaner and more concise is possible.
You dreamed [that] code that is cleaner and more concise is possible.
You dreamed [that] some degree of cleanliness and concision is possible.
Data classes enable you to write code that is cleaner and more concise than that.

Data classes enable you to write code that is [even] cleaner and more concise than [the code that's cleaner and more concise] [that] you ever dreamed [that] was possible.

Filling all these gaps with their inferred contents is a royal pain, and it leaves us with a sentence containing odd redundancies.  For a native writer, this is not a natural exercise and this is not a natural result.

We're explicitly comparing the cleanliness and concision of code using data classes to the cleanliness and concision of code that you dreamed was possible, and we're implicitly comparing the cleanliness and concision from that dream of a possibility with the cleanliness and concision of whatever code without data classes that you might already have used.

 
As to why "was possible" exists in the original, that's a question of style.  It doesn't have to.  It could vanish into its own gap.  It could be replaced by "possible" or "to be possible", which would change the grammatical analysis of the sentence but leave its overall semantics intact. 

The authors' intent is clear enough.  Not only will data classes allow you to write cleaner and more concise code than the code that you've written before, but even cleaner and more concise than code that you dreamed that it was possible to write.

Short answer: the subject is realized by a gap, which is linked to the noun phrase "code" in the matrix clause.

Longer answer:

I believe that this is the case of unbounded dependency construction defined by Huddleston & Pullum (2002) as:

An unbounded dependency construction is one which sanctions within it an anaphoric gap, with no upper bound on how deeply embedded the gap may be.

Let's take a look at the last sentence since it is shorter and easier to analyze:

That was definitely something _i she never dreamed ___ _i was possible

In this case, the subject is realized by a gap '____', which anaphorically linked to the noun phrase in the matrix clause "something".

Just to make it clearer, here's another example taken from Huddleston & Pullum (2002):

This is the man they think ___ attacked her

In this case, the subject in the subordinate clause is realized by a gap, which is then linked to the noun phrase "the man" in the matrix clause.

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EDIT: Your first and the main example is not an unbounded dependency construction. It is a comparative construction with a subject gap. This doesn't change the fact that the subject is realized by a gap, however.

I would not exactly call it an idiom.

Here, you’ll learn how data classes enable you to write code that’s cleaner and more concise than you ever dreamed was possible.

The writer assumes you dreamed of writing the cleanest code possible. The writer builds on the assumption that dreams and imagination can be pretty wild and exaggerated and informs you that things will be even better than you dreamed, i.e. better than you could imagine was possible. In other words, the writer promises to take you beyond what you imagined was possible in your wildest hopes and dreams.

I agree, it's a bit over the top and they could have ended with "dreams," but it has more emphasis this way. Sometimes people like to exaggerate a bit. And when reality breaks the bounds of our wildest dreams, hopes, and fantasies, well, we sometimes tend to use language like this. Hence the many "similar sentences" you found.

Just a word of warning regarding ads that sound too good to be true. Especially if you are not used to such "gushy" language, you might take it more literally than you should. Feel free to mentally drop the "was possible." After all, the writer has no way of knowing your personal dreams and what you dreamed was possible.

There are several things to discuss in this question, so bear with me as I go through them all.

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The referent of the dream is already identified in the example sentences:

• The code was cleaner than you ever dreamed was possible.
  → I never dreamed the code could be so clean.
• Life could be so much better than she had ever dreamed was possible.
  → I never dreamed life could be so good.
• You will know more about gold mining than you ever dreamed was possible.
  → I never dreamed I would know so much about gold mining.
• That was something she never dreamed was possible.
  → I never dreamed that was possible.

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Was possible acts in the same way as it normally does, but uses different wording when never is removed:

• I thought that cleanness of code was impossible.
• I thought that quality of life was impossible.
• I thought that level of knowledge about gold mining was impossible.
• I thought that was impossible.

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If was possible is omitted from the original sentences, then they are only saying you never dreamed about something, rather than discussing the possibility of something.

Consider the difference in nuance between the following two sentences:

• I never dreamed about living forever.
  → I never had a dream about living forever.
• I never dreamed living for every was possible.
  → I never thought it was possible to live forever.

With the use of was possible, the word dream changes from its sense of having visions while sleeping to having beliefs.

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The following is one sense of Merriam Webster's definition of dream:

1 : to have a series of thoughts, images, or emotions while sleeping : to have a dream …
    // doesn't recall dreaming last night
    // dream of departed loved ones

But the sense changes when was possible is part of the sentence:

2 : to consider as a possibility : IMAGINE
  // never dreamed I would become a teacher

The last example sentence Merriam-Webster uses is ambiguous on its own (without context), but it wouldn't be if it were written differently:

• I never dreamed becoming a teacher was possible.
  → I imagined (thought) becoming a teacher was impossible.