The closer we study life, the more extreme places we find it thriving.
Is there a proposition "at" missing at the end of this sentence above, to express the places at which life thrives?
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The sentence is unambiguous and comprehensible, and I can easily imagine a native speaker saying the sentence, but debatably it's incorrect. It's not obviously wrong; it's just perhaps a bit sloppy.
Informally, I would be happy that the following is correct:
The closer we study life, the more extreme the places we find it thriving.
This is to make the sentence match better the pattern "the more, the merrier", or "the more people, the merrier they are". In your original, it was more like "the more people, the merrier are". Your original is not as obviously weird as "the more people, the merrier are", but it is perhaps a little odd.
You could add the word "in" at the end of the sentence. I can't really articulate why I prefer "in" to "at", but I think "at" sounds non-native. The best I can offer is NGrams, where "in this place" is much more popular than "at this place".
In a formal context where I'm trying to be correct, I would say:
The more closely we study life, the more extreme are the places in which we find it thriving.
This is for several reasons:
By the way, you asked: "Are this sentence grammatically correct?". The subject of the sentence, namely the word "sentence", is singular, so the appropriate present-tense third-person form of "to be" is the singular "is" rather than the plural "are": "Is this sentence grammatically correct?".