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I have looked through several references, including Doug West's and Bela Bollobas's graph theory books, but I cannot find a proof of Brooks' Theorem that does not use the notion of blocks in a connected graph (its maximal 2-connected subgraphs). If anyone knows of a proof, I would greatly apreciate a reference. Thank you.

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    Please be kind to the answerers. What is Brooks's theorem? (I think it's Brooks, not Brook, but I still don't remember the theorem.) – 2017-01-01
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    https://en.m.wikipedia.org/wiki/Brooks'_theorem#Formal_statement – 2017-01-01

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Check out the two proofs in the book A Course in Combinatorics by Lint and Wilson, neither uses blocks.

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    Not quite true: the second proof uses the result that a simple graph with maximum degree $\le d$ that cannot be $d$-colored and is minimal with respect to these properties is $2$-connected, i.e., that it *is* a block. The first, though, definitely doesn’t use blocks. – 2017-01-02
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    @brianm.scott Oh I didnt motive that, thanks Brian. – 2017-01-02
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There are a few, this one is probably the simplest: https://arxiv.org/abs/1409.6812

For more proofs, see https://arxiv.org/abs/1403.0479

There are about ten proofs of Brooks theorem there. The proofs is sections 2,3,5,6,7 don't use blocks.